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    "# EART97012  <a class=\"tocSkip\">\n",
    "    \n",
    "# Geophysical Inversion <a class=\"tocSkip\">    \n",
    "    \n",
    "## Lecture 4 <a class=\"tocSkip\">   "
   ]
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    "### Today's learning objectives  <a class=\"tocSkip\">\n",
    "   \n",
    "     \n",
    "    \n",
    "1. To continue reviewing key linear algebra results, especially on the solvability of (square) linear systems\n",
    "    \n",
    "\n",
    "    \n",
    "2. To think about matrices and linear systems geometrically in terms of mappings between spaces (perhaps of different dimension, or where the mapping restricts the results to a lower-dimensional subspace)\n",
    "    \n",
    "    \n",
    "    \n",
    "3. To introduce the rank, range and null-space of matrices and to link these concepts to 1 and 2 above\n",
    "\n",
    "\n",
    "    \n",
    "4. To demonstrate how row operations can be used to establish rank and to find the null-space for a matrix\n",
    "    \n",
    "    \n",
    "    \n",
    "5. To introduce some simple examples of over- and under-determined problems and how the above concepts generalise, or are value, in the case of non-square matrices\n",
    "    \n",
    "\n",
    "6. To introduce the concepts of equi-, under-, over-, and mixed-determined problems"
   ]
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    "### Visualising matrix-vector multiplication in 3D  <a class=\"tocSkip\">\n",
    "\n",
    "Note that in order to consider non-trivial, non-square cases we really need to consider at least one of $m$ or $n$ to be at least 3.\n",
    "    \n",
    "This makes visualisation and interrogation of mappings between spaces a bit difficult - adding an ability to rotate 3D plots helps a little here.\n",
    "    \n",
    "There is a notebook in this directory that uses a different plotting backend and thus allows for the rotation of some of the mappings we consider in this lecture - take a look and try rotating the plots."
   ]
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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Introductory-comments\" data-toc-modified-id=\"Introductory-comments-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Introductory comments</a></span></li><li><span><a href=\"#Linear-(or-matrix)-systems-(a-reminder-on-square-systems)---continued\" data-toc-modified-id=\"Linear-(or-matrix)-systems-(a-reminder-on-square-systems)---continued-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Linear (or matrix) systems (a reminder on square systems) - continued</a></span><ul class=\"toc-item\"><li><span><a href=\"#Solvability-of-linear-systems\" data-toc-modified-id=\"Solvability-of-linear-systems-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Solvability of linear systems</a></span><ul class=\"toc-item\"><li><span><a href=\"#Linear-(in)dependence\" data-toc-modified-id=\"Linear-(in)dependence-2.1.1\"><span class=\"toc-item-num\">2.1.1&nbsp;&nbsp;</span>Linear (in)dependence</a></span></li><li><span><a href=\"#A-basis-for-a-vector-space\" data-toc-modified-id=\"A-basis-for-a-vector-space-2.1.2\"><span class=\"toc-item-num\">2.1.2&nbsp;&nbsp;</span>A basis for a vector space</a></span></li><li><span><a href=\"#An-example-|-the-geometrical-interpretation-of-matrix-vector-multiplication\" data-toc-modified-id=\"An-example-|-the-geometrical-interpretation-of-matrix-vector-multiplication-2.1.3\"><span class=\"toc-item-num\">2.1.3&nbsp;&nbsp;</span>An example | the geometrical interpretation of matrix vector multiplication</a></span></li></ul></li><li><span><a href=\"#Solvability-theory---some-equivalent-properties\" data-toc-modified-id=\"Solvability-theory---some-equivalent-properties-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Solvability theory - some equivalent properties</a></span></li><li><span><a href=\"#Rank,-range-and-null-space\" data-toc-modified-id=\"Rank,-range-and-null-space-2.3\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Rank, range and null-space</a></span><ul class=\"toc-item\"><li><span><a href=\"#Rank-and-range-for-square-systems\" data-toc-modified-id=\"Rank-and-range-for-square-systems-2.3.1\"><span class=\"toc-item-num\">2.3.1&nbsp;&nbsp;</span>Rank and range for square systems</a></span></li><li><span><a href=\"#The-non-square-case\" data-toc-modified-id=\"The-non-square-case-2.3.2\"><span class=\"toc-item-num\">2.3.2&nbsp;&nbsp;</span>The non-square case</a></span></li><li><span><a href=\"#The-null-space\" data-toc-modified-id=\"The-null-space-2.3.3\"><span class=\"toc-item-num\">2.3.3&nbsp;&nbsp;</span>The null-space</a></span></li><li><span><a href=\"#The-rank-nullity-theorem\" data-toc-modified-id=\"The-rank-nullity-theorem-2.3.4\"><span class=\"toc-item-num\">2.3.4&nbsp;&nbsp;</span>The rank-nullity theorem</a></span></li></ul></li><li><span><a href=\"#(Reduced)-row-echelon-form-of-a-matrix-and-matrix-rank-and-null-space\" data-toc-modified-id=\"(Reduced)-row-echelon-form-of-a-matrix-and-matrix-rank-and-null-space-2.4\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>(Reduced) row echelon form of a matrix and matrix rank and null space</a></span></li></ul></li><li><span><a href=\"#A-first-look-at-some-simple-non-square-systems\" data-toc-modified-id=\"A-first-look-at-some-simple-non-square-systems-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>A first look at some simple non-square systems</a></span><ul class=\"toc-item\"><li><span><a href=\"#A-simple-over-determined-case\" data-toc-modified-id=\"A-simple-over-determined-case-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>A simple over-determined case</a></span></li><li><span><a href=\"#A-simple-under-determined-case\" data-toc-modified-id=\"A-simple-under-determined-case-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>A simple under-determined case</a></span><ul class=\"toc-item\"><li><span><a href=\"#An-under-determined-problem-with-no-solutions,-and-with-infinitely-many-solutions\" data-toc-modified-id=\"An-under-determined-problem-with-no-solutions,-and-with-infinitely-many-solutions-3.2.1\"><span class=\"toc-item-num\">3.2.1&nbsp;&nbsp;</span>An under-determined problem with no solutions, and with infinitely many solutions</a></span></li></ul></li><li><span><a href=\"#A-summary-of-what-we've-establised-so-far-about-solution-methods\" data-toc-modified-id=\"A-summary-of-what-we've-establised-so-far-about-solution-methods-3.3\"><span class=\"toc-item-num\">3.3&nbsp;&nbsp;</span>A summary of what we've establised so far about solution methods</a></span></li></ul></li><li><span><a href=\"#Over-,-under-,-equi--and-mixed-determined-problems\" data-toc-modified-id=\"Over-,-under-,-equi--and-mixed-determined-problems-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Over-, under-, equi- and mixed-determined problems</a></span><ul class=\"toc-item\"><li><span><a href=\"#Introduction\" data-toc-modified-id=\"Introduction-4.1\"><span class=\"toc-item-num\">4.1&nbsp;&nbsp;</span>Introduction</a></span><ul class=\"toc-item\"><li><span><a href=\"#The-equi-determined-case\" data-toc-modified-id=\"The-equi-determined-case-4.1.1\"><span class=\"toc-item-num\">4.1.1&nbsp;&nbsp;</span>The equi-determined case</a></span></li><li><span><a href=\"#Aside---the-Rouché–Capelli-theorem\" data-toc-modified-id=\"Aside---the-Rouché–Capelli-theorem-4.1.2\"><span class=\"toc-item-num\">4.1.2&nbsp;&nbsp;</span>Aside - the Rouché–Capelli theorem</a></span></li><li><span><a href=\"#The-over-determined-case\" data-toc-modified-id=\"The-over-determined-case-4.1.3\"><span class=\"toc-item-num\">4.1.3&nbsp;&nbsp;</span>The over-determined case</a></span></li><li><span><a href=\"#The-under-determined-case\" data-toc-modified-id=\"The-under-determined-case-4.1.4\"><span class=\"toc-item-num\">4.1.4&nbsp;&nbsp;</span>The under-determined case</a></span></li><li><span><a href=\"#The-mixed-determined-case\" data-toc-modified-id=\"The-mixed-determined-case-4.1.5\"><span class=\"toc-item-num\">4.1.5&nbsp;&nbsp;</span>The mixed-determined case</a></span></li></ul></li><li><span><a href=\"#Matrix-rank-in-the-non-square-case\" data-toc-modified-id=\"Matrix-rank-in-the-non-square-case-4.2\"><span class=\"toc-item-num\">4.2&nbsp;&nbsp;</span>Matrix rank in the non-square case</a></span><ul class=\"toc-item\"><li><span><a href=\"#Example:-matrix-(outer)-products\" data-toc-modified-id=\"Example:-matrix-(outer)-products-4.2.1\"><span class=\"toc-item-num\">4.2.1&nbsp;&nbsp;</span>Example: matrix (outer) products</a></span></li></ul></li><li><span><a href=\"#Some-examples\" data-toc-modified-id=\"Some-examples-4.3\"><span class=\"toc-item-num\">4.3&nbsp;&nbsp;</span>Some examples</a></span><ul class=\"toc-item\"><li><span><a href=\"#Example-1\" data-toc-modified-id=\"Example-1-4.3.1\"><span class=\"toc-item-num\">4.3.1&nbsp;&nbsp;</span>Example 1</a></span></li><li><span><a href=\"#Example-2\" data-toc-modified-id=\"Example-2-4.3.2\"><span class=\"toc-item-num\">4.3.2&nbsp;&nbsp;</span>Example 2</a></span></li><li><span><a href=\"#Example-3---A-tomography-inspired-mixed-determined-problem\" data-toc-modified-id=\"Example-3---A-tomography-inspired-mixed-determined-problem-4.3.3\"><span class=\"toc-item-num\">4.3.3&nbsp;&nbsp;</span>Example 3 - A tomography inspired mixed-determined problem</a></span></li><li><span><a href=\"#The-issue-of-imperfect-data\" data-toc-modified-id=\"The-issue-of-imperfect-data-4.3.4\"><span class=\"toc-item-num\">4.3.4&nbsp;&nbsp;</span>The issue of imperfect data</a></span></li></ul></li></ul></li><li><span><a href=\"#Summary\" data-toc-modified-id=\"Summary-5\"><span class=\"toc-item-num\">5&nbsp;&nbsp;</span>Summary</a></span></li></ul></div>"
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    "%matplotlib inline\n",
    "%precision 6\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import scipy.linalg as sl\n",
    "from pprint import pprint"
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    "# Introductory comments \n",
    "\n",
    "We'll being today by wrapping up our discussion of the solvability of square systems before moving on to some simple examples of non-square systems."
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    "# Linear (or matrix) systems (a reminder on square systems) - continued\n",
    "\n",
    "Consider the problem written in the form\n",
    "\n",
    "$$A\\boldsymbol{x}=\\boldsymbol{b},$$\n",
    "\n",
    "where for a given $m\\times n$ matrix $A$ and a given $m\\times 1$ right hand side vector $\\boldsymbol{b}\\;$ we want to find an $n\\times 1$ vector $\\boldsymbol{x}\\;$.\n",
    "\n",
    "We'll continue thinking about the square problem, i.e. where $m=n$, before moving on to the case on non-square problems we are particularly interested in in this module, i.e. where the vectors $\\boldsymbol{x}$ and $\\boldsymbol{b}$ are of different lengths, or where we can interpret multiplication by $A$ as mapping vectors from $\\mathbb{R}^n$ into $\\mathbb{R}^m$.\n",
    "\n",
    "In all situations it's important that we understand **whether our problem has a solution** and **whether that solution is unique**."
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    "## Solvability of linear systems\n",
    "\n",
    "\n",
    "If $\\det(A)=0$ ($A$ is singular) a linear system featuring the matrix $A$ with any RHS does **not** have a **unique** solution, it **may have either infinite *or* no solutions**.\n",
    "\n",
    "For example, consider\n",
    "\n",
    "$$\n",
    "\\left(\n",
    "  \\begin{array}{rr}\n",
    "    2 & 3 \\\\\n",
    "    4 & 6 \n",
    "  \\end{array}\n",
    "\\right)\\left(\n",
    "  \\begin{array}{c}\n",
    "    x \\\\\n",
    "    y\n",
    "  \\end{array}\n",
    "\\right) = \\left(\n",
    "  \\begin{array}{c}\n",
    "    4 \\\\\n",
    "    8\n",
    "  \\end{array}\n",
    "\\right),\n",
    "$$\n",
    "\n",
    "where the matrix on the LHS clearly has a zero determinant.\n",
    "\n",
    "Considering now the values in the RHS vector as well, the second equation is simply twice the first, and hence a solution to the first equation is also automatically a solution to the second equation.  In this case as the equations do not contradict one another we say that the equations are **consistent**.\n",
    "\n",
    "If we think geometrically, and interpret the two equations as constraints, they are **both** constraining our $x,y$ values to the same 1D subspace - any solution along this line in 2D satisfies **both** equations, and hence any of them is a solution to our linear system.\n",
    "\n",
    "We hence only have one *linearly-independent* equation here, and our problem is under-constrained: we effectively only have one equation for two unknowns and this problem has *infinitely many* possibly solutions (or said another way, we have *existence*, but *non-uniqueness*): e.g. $\\boldsymbol{x}=(2,0)^T$ is a solution, so is $\\boldsymbol{x}=(-1,2)^T$, etc.\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "If we instead replaced the RHS vector with $(4,7)^T$, then the two equations would now be contradictory: in this case we have *no solutions* (or *non-existence*).\n",
    "\n",
    "<br>\n",
    "\n",
    "This was an example in last week's homework exercises - please review."
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    "### Linear (in)dependence\n",
    "\n",
    "Note that a set of vectors where one can be written as a linear sum of the others are termed [***linearly-dependent***](https://en.wikipedia.org/wiki/Linear_independence). \n",
    "\n",
    "(As was the case for the rows and the columns in the example immediately above - the first row of $A$ is simply twice the second; the second column is $3/2$ times the first).\n",
    "\n",
    "<br>\n",
    "\n",
    "When this is **not** the case the vectors are termed [***linearly-independent***](https://en.wikipedia.org/wiki/Linear_independence).\n",
    "\n",
    "<br>\n",
    "\n",
    "When the rows are linearly dependent what can you conclude about the expected structure of the row echelen form of the matrix? I.e. what do you expect to end up if you add and subtract multiples of rows to other rows in order to simplify the matrix?\n"
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    "### A basis for a vector space\n",
    "\n",
    "A <a href=\"https://en.wikipedia.org/wiki/Basis_(linear_algebra)\">*basis*</a> for a vector (or linear) space is a set of *linearly-independent* vectors such that every vector (or location) in the vector space is a linear combination of this set.\n",
    "\n",
    "[This a [goldilocks](https://en.wikipedia.org/wiki/Goldilocks_and_the_Three_Bears) like definition - we have exactly the right number of vectors - too few and we can't reach every point in the space through their combination, and too many some will be redundant and will be linearly dependent.]\n",
    "\n",
    "The number of vectors in this set is called the **dimension** of the vector space.\n",
    "\n",
    "For example, the unit vectors $\\,\\boldsymbol{i}$, $\\boldsymbol{j}$, $\\boldsymbol{k}\\,$ are linearly-independent and form a basis for the [three-dimensional space $\\mathbb{R}^3$](https://en.wikipedia.org/wiki/Three-dimensional_space). These unit vectors are defined as\n",
    "\n",
    "$$\\boldsymbol{i} = \\left(\n",
    "  \\begin{array}{c}\n",
    "    1 \\\\\n",
    "    0 \\\\\n",
    "    0     \n",
    "  \\end{array}\n",
    "\\right),\n",
    "\\;\\;\\;\\;\\;\\;\\;\\;\n",
    "\\boldsymbol{j} = \\left(\n",
    "  \\begin{array}{c}\n",
    "    0 \\\\\n",
    "    1 \\\\\n",
    "    0     \n",
    "  \\end{array}\n",
    "\\right),\n",
    "\\;\\;\\;\\;\\;\\;\\;\\;\n",
    "\\boldsymbol{k} = \\left(\n",
    "  \\begin{array}{c}\n",
    "    0 \\\\\n",
    "    0 \\\\\n",
    "    1     \n",
    "  \\end{array}\n",
    "\\right).\n",
    "$$\n",
    "\n",
    "By definition, these three vectors **forming a basis means that we can reach any point in $\\mathbb{R}^3$ through summing appropriate multiples of them**.\n",
    "\n",
    "These are the basis vectors which we commonly choose (as they're in some sense the simplest and cleanest choice), but clearly they are not unique - with any linearly independent set of three vectors we can reach any point in 3D space through their weighted summation - start thinking about these basis vectors being columns of a matrix forming a linear system!"
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    "### An example | the geometrical interpretation of matrix vector multiplication\n",
    "\n",
    "Consider our $2\\times 2$ example from earlier\n",
    "\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "  2x + 3y &= 7 \\\\[5pt]\n",
    "   x - 4y &= 3 \n",
    "\\end{align*}\n",
    "   \\quad \\iff \\quad\n",
    "  \\begin{pmatrix}\n",
    "    2 & 3 \\\\\n",
    "    1 & -4  \n",
    "  \\end{pmatrix}\n",
    "  \\begin{pmatrix}\n",
    "    x \\\\\n",
    "    y \n",
    "  \\end{pmatrix}=\n",
    "  \\begin{pmatrix}\n",
    "    7 \\\\\n",
    "    3 \n",
    "  \\end{pmatrix}   \n",
    "$$\n",
    "\n",
    "If we define two vectors given by the columns of the matrix, which we plot below in blue and red, \n",
    "\n",
    "along with the point $\\boldsymbol{b}$ we want to arrive at,\n",
    "\n",
    "\n",
    "we can then visualise the problem of solving the linear system as wanting to reach the black point through the appropriate weighted sum of the blue and red vectors.  Why is this?"
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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(6, 6))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.set_title('Position as a sum over matrix cols', fontsize=14)\n",
    "#ax1.grid(True)\n",
    "\n",
    "i = np.array([1,0])\n",
    "j = np.array([0,1])\n",
    "\n",
    "# plot the unit vectors\n",
    "ax1.quiver(i[0], i[1], angles='xy', scale_units='xy', scale=1, color='k')\n",
    "ax1.quiver(j[0], j[1], angles='xy', scale_units='xy', scale=1, color='k')\n",
    "\n",
    "ax1.set_xlim(-1,8)\n",
    "ax1.set_ylim(-5,5)\n",
    "\n",
    "A = np.array([[2, 3], [1, -4]])\n",
    "vec1 = A[:,0] # NB. This is the same as A*x\n",
    "vec2 = A[:,1] # NB. This is the same as A*y\n",
    "\n",
    "# plot them\n",
    "ax1.quiver(vec1[0], vec1[1], angles='xy', scale_units='xy', scale=1, color='b', width=0.02,  label='vec1')\n",
    "ax1.quiver(vec2[0], vec2[1], angles='xy', scale_units='xy', scale=1, color='r', width=0.02,  label='vec2')\n",
    "\n",
    "# add point b\n",
    "b = np.array([7., 3.])\n",
    "ax1.plot(b[0], b[1], 'ko',  label='${b}$')\n",
    "\n",
    "ax1.legend(loc='lower left', fontsize=14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As already stated, we know trivially how to write the position $\\boldsymbol{b}$ as a sum over the unit vectors $\\boldsymbol{i}$ and $\\boldsymbol{j}$ (the black vectors indicate the unit vectors $\\boldsymbol{i}$ and $\\boldsymbol{j}$ in the plot above):\n",
    "\n",
    "$$\\boldsymbol{b} = \n",
    "\\begin{pmatrix}\n",
    "b_1\\\\\n",
    "b_2\n",
    "\\end{pmatrix} =\n",
    "\\begin{pmatrix}\n",
    "b_1\\\\\n",
    "0\n",
    "\\end{pmatrix} +\n",
    "\\begin{pmatrix}\n",
    "0\\\\\n",
    "b_2\n",
    "\\end{pmatrix} =\n",
    "b_1\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\n",
    "\\end{pmatrix} + b_2\n",
    "\\begin{pmatrix}\n",
    "0\\\\\n",
    "1\n",
    "\\end{pmatrix} =\n",
    "b_1 \\boldsymbol{i} + b_2\\boldsymbol{j}\n",
    "$$\n",
    "\n",
    "Finding the solution to our linear system is equivalent to asking how far along direction given by column one (blue vector), and then how far along the direction given by column two (the red vector), do I need to go to arrive at point $\\boldsymbol{b}$.\n",
    "\n",
    "This is exactly what the following interpretation of matrix-vector multiplication from earlier is telling us\n",
    "\n",
    "$$ A \\boldsymbol{x}=\\boldsymbol{b} \\quad \\iff \\quad\n",
    "\\begin{pmatrix}\n",
    "    b_1\\\\\n",
    "    b_2\\\\\n",
    "    \\vdots\\\\\n",
    "    b_m\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "      &     &         &     \\\\\n",
    "      &     &         &     \\\\      \n",
    "  \\boldsymbol{a}_{\\,:1} & \\boldsymbol{a}_{\\,:2} &  \\ldots & \\boldsymbol{a}_{\\,:n} \\\\\n",
    "      &     &         &     \\\\\n",
    "      &     &         &     \n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "    x_1\\\\\n",
    "    x_2\\\\\n",
    "    \\vdots\\\\\n",
    "    x_n\n",
    "\\end{pmatrix}\n",
    "=\n",
    "x_1 \n",
    "\\begin{pmatrix}\n",
    "       \\\\\n",
    "       \\\\\n",
    "   \\boldsymbol{a}_{\\,:1} \\\\\n",
    "       \\\\\n",
    "    ~   \n",
    "\\end{pmatrix}\n",
    "+\n",
    "x_2 \n",
    "\\begin{pmatrix}\n",
    "       \\\\\n",
    "       \\\\\n",
    "   \\boldsymbol{a}_{\\,:2} \\\\\n",
    "       \\\\\n",
    "    ~   \n",
    "\\end{pmatrix}\n",
    "+ \\cdots +\n",
    "x_n\n",
    "\\begin{pmatrix}\n",
    "       \\\\\n",
    "       \\\\\n",
    "   \\boldsymbol{a}_{\\,:n} \\\\\n",
    "       \\\\\n",
    "    ~   \n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "\n",
    "or in our case\n",
    "\n",
    "$$ \n",
    "\\begin{pmatrix}\n",
    "    7\\\\\n",
    "    3\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "      2 & 3   \\\\\n",
    "     1  & -4    \n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "    x_1\\\\\n",
    "    x_2\n",
    "\\end{pmatrix}\n",
    "=\n",
    "x_1 \n",
    "\\begin{pmatrix}\n",
    "       2 \\\\\n",
    "       1\n",
    "\\end{pmatrix}\n",
    "+\n",
    "x_2 \n",
    "\\begin{pmatrix}\n",
    "     3 \\\\\n",
    "     -4\n",
    "\\end{pmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the solution $\\boldsymbol{x}$ we have already found (`x = np.array([37./11, 1./11])`) to emphasise this visually/geometrically: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.quiver.Quiver at 0x138a6575708>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(8, 8))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.set_title('Position as a sum over matrix cols', fontsize=14)\n",
    "#ax1.grid(True)\n",
    "\n",
    "i = np.array([1,0])\n",
    "j = np.array([0,1])\n",
    "\n",
    "# plot the unit vectors\n",
    "ax1.quiver(i[0], i[1], angles='xy', scale_units='xy', scale=1, color='k')\n",
    "ax1.quiver(j[0], j[1], angles='xy', scale_units='xy', scale=1, color='k')\n",
    "\n",
    "\n",
    "ax1.set_xlim(-1,8)\n",
    "ax1.set_ylim(-5,5)\n",
    "\n",
    "A = np.array([[2, 3], [1, -4]])\n",
    "vec1 = A[:,0] # NB. This is the same as A*x\n",
    "vec2 = A[:,1] # NB. This is the same as A*y\n",
    "\n",
    "# plot them\n",
    "ax1.quiver(vec1[0], vec1[1], angles='xy', scale_units='xy', scale=1, color='b', width=0.02,  label='vec1')\n",
    "ax1.quiver(vec2[0], vec2[1], angles='xy', scale_units='xy', scale=1, color='r', width=0.02,  label='vec2')\n",
    "\n",
    "# add point b\n",
    "b = np.array([7., 3.])\n",
    "ax1.plot(b[0], b[1], 'ko',  label='${b}$')\n",
    "\n",
    "ax1.legend(loc='lower left', fontsize=14)\n",
    "\n",
    "# Now move x_1 in the direction given by column one, followed by x_2 in the direction given by column 2\n",
    "# and add to the plot with thinner lines\n",
    "x = np.array([37./11, 1./11])\n",
    "ax1.quiver(x[0]*vec1[0], x[0]*vec1[1], angles='xy', scale_units='xy', scale=1, color='darkblue', width=0.005,  label='vec1')\n",
    "ax1.quiver(x[0]*vec1[0], x[0]*vec1[1],x[1]*vec2[0], x[1]*vec2[1], angles='xy', \n",
    "           scale_units='xy', scale=1, color='darkred', width=0.005,  label='vec2')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we need to take 37/11 multiplied by the blue arrow and add 1/11 multiplied by the red arrow to reach the black dot.\n",
    "\n",
    "[Note we don't say something like \"move 37/11 in the direction of the blue arrow, as we might with i,j,k, since these vectors are not normalised\"].\n",
    "\n",
    "Note that the columns are not orthogonal in this case, which we can demonstrate by:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n"
     ]
    }
   ],
   "source": [
    "print(vec1.dot(vec2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "if they were orthogonal the answer should have been zero,\n",
    "\n",
    "but that's fine as we can still reach any position in 2D space from their weighted sum as long as they are linearly independent, i.e. in the 2D case as long as they don't point in the same direction."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solvability theory - some equivalent properties\n",
    "\n",
    "\n",
    "The following properties of a square $n\\times n$ matrix are equivalent:\n",
    "\n",
    "\n",
    "\n",
    "* $\\det(A)\\ne 0$, i.e. $A$ is non-singular.\n",
    "\n",
    "\n",
    "\n",
    "* The columns of $A$ are linearly independent.\n",
    "\n",
    "\n",
    "* The rows of $A$ are linearly independent.\n",
    "\n",
    "\n",
    "* The columns of $A$ [*span*](https://en.wikipedia.org/wiki/Linear_span) an $n$-dimensional space (this means that we can reach any point in $\\mathbb{R}^n$ through a linear combination of these vectors - recall we saw previously that this is simply what the operation $A\\boldsymbol{x}$ is doing if you write it out - the entries of $\\boldsymbol{x}$ are the weights in the linear combination - we wrote this out explicitly above.)  \n",
    "\n",
    "\n",
    "* [what this means is that for **any** given point in our space (described for example by a given $\\boldsymbol{b}$), we can reach that point through the appropriate summation of multiples of the columns of $A$, and its the $\\boldsymbol{x}$ solution vector that gives us these multiples.]\n",
    "\n",
    "\n",
    "* $A$ is invertible, i.e. there exists a matrix $A^{-1}$ such that $A^{-1}A = A A^{-1}=I$.\n",
    "\n",
    "\n",
    "* The matrix system $A\\boldsymbol{x}=\\boldsymbol{b}$ has a unique solution for every vector $\\boldsymbol{b}$.\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "For even more equivalent properties for square matrices see <https://en.wikipedia.org/wiki/Invertible_matrix#The_invertible_matrix_theorem>."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rank, range and null-space\n",
    "\n",
    "Above we wrote for a square $n\\times n$ matrix\n",
    "\n",
    "<br>\n",
    "\n",
    "* The columns of $A$ [*span*](https://en.wikipedia.org/wiki/Linear_span) an $n$-dimensional space (this means that we can reach any point in $\\mathbb{R}^n$ through a linear combination of these vectors - note that this is simply what the operation $A\\boldsymbol{x}$ is doing of course if you write it out - the entries of $\\boldsymbol{x}$ are the weights in the linear combination!)  \n",
    "\n",
    "<br>\n",
    "\n",
    "The columns of $A$ will always span an $r$-dimensional space, and assuming the matrix isn't the zero matrix $r \\ge 1$. If all the columns are linearly dependent then $r=1$. \n",
    "\n",
    "What we're saying in the statement above is that *if the columns of $A$ span an $n$-dimensional space* (i.e. if $r=n$) then ... all the other properties in the previous cell hold.\n",
    "\n",
    "<br>\n",
    "\n",
    "### Rank and range for square systems\n",
    "\n",
    "Let's explain a bit better what this means.\n",
    "\n",
    "The *rank* of an $n\\times n$ matrix $A$, written $\\text{rank}(A)$, is equal to the number of linearly independent columns that form it. It is also equal to the number of linearly independent rows that form it (as these two quantities, the column rank and the row rank can be proved to be equal). \n",
    "\n",
    "The matrix $A$ is said to have *full rank* if $\\text{rank}(A)=n$ and is said to be *rank deficient* if $\\text{rank}(A)<n)$ (in the square case - see below for the non-square case).\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "\n",
    "The *range* (sometimes the *image*) of an $n\\times n$ matrix $A$, written $\\text{range}(A)$, is the space spanned by the columns of $A$ (so it's also sometimes also called the [*column space*](https://en.wikipedia.org/wiki/Row_and_column_spaces)), i.e. every point in that space can be reached through a linear combination of the columns of $A$, equivalently every point in the space can be written as $A\\boldsymbol{x}$ for some $\\boldsymbol{x}$. Said another way, the columns of $A$ provide a *basis* for $\\text{range}(A)$.\n",
    "\n",
    "This is telling us that if our matrix has full rank then we can reach ***any*** location in $\\mathbb{R}^n$ through a weighted sum of the columns, i.e. no matter what the RHS vector is (the point we want to reach after we multiply by $A$) the corresponding linear system always has a unique solution and hence the matrix has an inverse. \n",
    "\n",
    "<br>\n",
    "\n",
    "If the matrix does not have full rank (i.e. $r<n$), that means we cannot reach all point in $\\mathbb{R}^n$ through a weighted sum of the columns. We will have (**non-unique**) solutions as long as the RHS vector (the location we want to get to geometrically) is in the range of $A$, as we are able to reach some points (i.e. some RHS vectors) with a weighted sum of our columns. \n",
    "\n",
    "If our given RHS vector is not in the range of $A$ then we simply cannot reach it through the weighted summation of columns (this is the **non-existence** case).\n",
    "\n",
    "Note that the difference between the last two points is completely down to the RHS vector and is nothing to do with the entries of the matrix $A$ - see the example from the start.\n",
    "\n",
    "<br>\n",
    "\n",
    "### The non-square case\n",
    "\n",
    "Note that much of this generalises to the non-square case, as we'll see towards the end of this lecture."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The null-space\n",
    "\n",
    "The *null-space* of the matrix $A$, written $\\text{null}(A)$, is the set of vectors $\\boldsymbol{x}$ that satisfy $A\\boldsymbol{x} = 0$.\n",
    "\n",
    "Why is the null-space important, well for lots of reasons including: \n",
    "\n",
    "\n",
    "1. in the next lecture it is how we will find so-called eigenvectors, \n",
    "    \n",
    "    \n",
    "    \n",
    "2. when solving a linear system $A\\boldsymbol{x}=\\boldsymbol{b}$, we can add any vector (or linear combination of vectors) from the null-space to a solution and get another solution, this will demonstrate non-uniqueness of solutions for this problem if the null-space contains anything other than the zero vector. \n",
    "\n",
    "\n",
    "Note that the null-space always contains the zero vector, we are not really interested in that solution, it's non-zero vectors that are important. Note if we have a non-zero vector in the null-space, then any scalar multiple of that vector will also be in the null space ($A\\boldsymbol{x}=\\boldsymbol{0}\\Rightarrow A(\\alpha\\boldsymbol{x})=\\alpha A \\boldsymbol{x}= \\boldsymbol{0}$).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### The rank-nullity theorem\n",
    "\n",
    "A fundamental result for any matrix $A \\in \\mathbb{R}^{m\\times n}$ (i.e. not necessarily square) which maps vectors from $\\mathbb{R}^{n}$ into $\\mathbb{R}^{m}$, is the so-called [rank-nullity theorem](https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem).\n",
    "\n",
    "This states that the dimension of the span of the columns of $A$ (i.e. its rank) and the dimension of its null space (termed the nullity) must sum to $n$ - the number of columns:\n",
    "\n",
    "$$ \\text{rank}(A) + \\dim(\\text{null}(A)) = n$$\n",
    "\n",
    "<br>\n",
    "\n",
    "This means that if you have worked out the rank you automatically know the dimension of the null space, in particular if it contains any non-zero vectors $\\boldsymbol{x}$ such that $A\\boldsymbol{x}=\\boldsymbol{0}$.\n",
    "\n",
    "And vice-versa, if you know that you have a non-zero vector in the null space (i.e. its dimension is at least 1) then the matrix cannot be full rank.\n",
    "\n",
    "<br>\n",
    "\n",
    "Note that this will always be the case for  under-determined systems where $m<n$ as the rank can be no larger than  $\\min(m,n)$.\n",
    "\n",
    "It may be the case for over-determined systems."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## (Reduced) row echelon form of a matrix and matrix rank and null space\n",
    "\n",
    "\n",
    "Recall from the last lecture that we noted that row operations do not change the solution to a linear system as long as we apply the same operations to the RHS vector. \n",
    "\n",
    "\n",
    "Recall that **row operations** are defined as\n",
    "\n",
    "1. multiply each row by a non-zero scalar\n",
    "\n",
    "\n",
    "2. add multiples of one row to another \n",
    "\n",
    "\n",
    "3. swap rows\n",
    "\n",
    "<br>\n",
    "\n",
    "It's *also* the case that row operations do not change the rank of a matrix. Therefore one way to establish the rank of a matrix is to transform it into (R)REF and read off the rank of the transformed matrix as the number of linearly independent rows (equivalently the number of linearly independent columns), and this is also the rank of the matrix in its original form.  \n",
    "\n",
    "\n",
    "Note that sometimes we will want to compute the rank of a matrix $A$, but sometimes the rank of the augmented matrix $[A|\\boldsymbol{b}]$.\n",
    "\n",
    "<br>\n",
    "\n",
    "Note that this also yields a method to compute the null space - we say this in the section last week \"Problems with a zero RHS\", but will go through some more examples today.\n",
    "    \n",
    "<br> \n",
    "\n",
    "Recall that a matrix is in [*row echelon form*](https://en.wikipedia.org/wiki/Row_echelon_form) when\n",
    "\n",
    "\n",
    "- all non-zero rows are above any rows of all zeros (NB. if we do have rows of all zeros then the matrix is singular), and\n",
    "\n",
    "\n",
    "- the leading coefficient in the row (first non-zero entry from the left) is always strictly to the right of the leading coefficient of the row above\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "- If in addition the leading coefficient in any non-zero row is unity and every column containing a leading '1' has zeros everywhere else within that column, we say the matrix is in *reduced row echelon form*.\n",
    "\n",
    "\n",
    "\n",
    "<br><br><br>\n",
    "\n",
    "Please have a read through of the homework exercise (given in the L2 homework notebook) that goes through the steps required to use row operations in order to compute the rank of the matrix and to compute the corresponsing null space. It also includes a visualisation of the transformation that results from multiplication by the matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A first look at some simple non-square systems\n",
    "\n",
    "So hopefully now we have a good overview of the square case where the number of unknowns is equal to the number of equations. Let's think about how things change when these two numbers are not balanced.\n",
    "\n",
    "\n",
    "## A simple over-determined case\n",
    "\n",
    "Let's start from our square system from earlier\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "  2x + 3y &= 7 \\\\[5pt]\n",
    "   x - 4y &= 3 \n",
    "\\end{align*}\n",
    "   \\quad \\iff \\quad\n",
    "  \\begin{pmatrix}\n",
    "    2 & 3 \\\\\n",
    "    1 & -4  \n",
    "  \\end{pmatrix}\n",
    "  \\begin{pmatrix}\n",
    "    x \\\\\n",
    "    y \n",
    "  \\end{pmatrix}=\n",
    "  \\begin{pmatrix}\n",
    "    7 \\\\\n",
    "    3 \n",
    "  \\end{pmatrix}   \n",
    "$$\n",
    "\n",
    "We rearranged each to the following in order to easily draw two straight lines:\n",
    "\n",
    "\\begin{align*}\n",
    "  y &= -\\frac{2}{3}x + \\frac{7}{3} \\\\[5pt]\n",
    "  y &= \\frac{1}{4}x - \\frac{3}{4}\n",
    "\\end{align*}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<matplotlib.legend.Legend at 0x138a6528e08>"
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     "execution_count": 5,
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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0,5,100)\n",
    "\n",
    "y1 = -(2./3.)*x + (7./3.)\n",
    "y2 = (1./4.)*x - (3./4.)\n",
    "\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.grid(True)\n",
    "\n",
    "ax1.plot(x,y1,'b', label='$2x+3y=7$')\n",
    "ax1.plot(x,y2,'r', label='$x-4y=3$')\n",
    "ax1.plot(37./11,1./11, 'ko', label='Our solution')\n",
    "\n",
    "ax1.legend(loc='best', fontsize=14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The unique solution to this so-called **equi-determined** case is given by the black dot - the $(x,y)$ values that satisfy both equations/constraints.  \n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "Now let's now suppose we impose an additional constraint, or ask that our solution solves an additional equation. We call this an ***over-determined*** or an over-constrained problem - we have more equations/constraints than unknowns.\n",
    "\n",
    "What would this look like visually/geometrically?\n",
    "\n",
    "<br>\n",
    "\n",
    "A third line (corresponding to our third equation) would be added to our plot and we have three possible situations we need to consider:\n",
    "\n",
    "\n",
    "1. the new line doesn't pass through the black dot\n",
    "\n",
    "2a. the new line does go through the black dot and **is not** coincident with either of the first two\n",
    "\n",
    "[2b. the new line does go through the black dot and **is** coincident with either of the first two]\n",
    "\n",
    "<br>\n",
    "\n",
    "Let's see examples of the first two situations - the third can be ignored as it effectively reduces back to the original two equation / two unknown case (the new constraint isn't actually new at all, we've just repeated an equation - both LHS and RHS)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x138a6827648>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# a carefully chosen new constraint/equation/line that does go through our existing inersection/solution\n",
    "\n",
    "x = np.linspace(0,5,100)\n",
    "\n",
    "y1 = -(2./3.)*x + (7./3.)\n",
    "y2 = (1./4.)*x - (3./4.)\n",
    "y3 = -(3./10.)*x + (11./10.)\n",
    "\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.grid(True)\n",
    "\n",
    "ax1.plot(x,y1,'b', label='$2x+3y=7$')\n",
    "ax1.plot(x,y2,'r', label='$x-4y=3$')\n",
    "ax1.plot(x,y3,'g', label='$-3x-10y=-11$')\n",
    "ax1.plot(37./11,1./11, 'ko', label='Our solution')\n",
    "\n",
    "ax1.legend(loc='best', fontsize=14)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our system of simultaneous equations is now\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "  2x + 3y &= 7 \\\\[5pt]\n",
    "   x - 4y &= 3 \\\\[5pt]\n",
    "  -3x - 10y & = -11\n",
    "\\end{align*}\n",
    "   \\quad \\iff \\quad\n",
    "  \\begin{pmatrix}\n",
    "    2 & 3 \\\\\n",
    "    1 & -4  \\\\\n",
    "    -3 & -10 \n",
    "  \\end{pmatrix}\n",
    "  \\begin{pmatrix}\n",
    "    x \\\\\n",
    "    y \n",
    "  \\end{pmatrix}=\n",
    "  \\begin{pmatrix}\n",
    "    7 \\\\\n",
    "    3 \\\\\n",
    "    -11\n",
    "  \\end{pmatrix}   \n",
    "$$\n",
    "\n",
    "We rearranged each to the following in order to easily draw three straight lines:\n",
    "\n",
    "\\begin{align*}\n",
    "  y &= -\\frac{2}{3}x + \\frac{7}{3} \\\\[5pt]\n",
    "  y &= \\frac{1}{4}x - \\frac{3}{4} \\\\[5pt]\n",
    "  y &= -\\frac{3}{10}x + \\frac{11}{10}\n",
    "\\end{align*}\n",
    "\n",
    "<br>\n",
    "\n",
    "So even though we have an additional equation/constraint, and our system is not square (it is over-determined as we have more equations than unknowns), we still have a unique solution.\n",
    "\n",
    "This won't generally be the case with an over-determined problem, and only arose here as I was careful to choose a third equation that didn't actually impose any new information (note that it's just a linear combination of the other two).\n",
    "\n",
    "<br>\n",
    "\n",
    "We can plot the transformation\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  7.   3. -11.]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# let's plot the transformation\n",
    "\n",
    "A = np.array([[2, 3], [1, -4], [-3, -10]])\n",
    "\n",
    "x = np.linspace(-5, 5, 20)\n",
    "y = np.linspace(-5, 5, 20)\n",
    "# this creates a mesh of points in 2D\n",
    "xx, yy = np.meshgrid(x, y)\n",
    "# convert to row vectors\n",
    "xxx = np.reshape(xx,(1,np.size(xx)))\n",
    "yyy = np.reshape(yy,(1,np.size(yy)))\n",
    "# convert to a 2 x N matrix of vectors/points\n",
    "vecs = np.vstack((xxx,yyy))\n",
    "\n",
    "# transform these points\n",
    "Avecs = A@vecs\n",
    "\n",
    "fig = plt.figure(figsize=(12, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(1, 2, 1)\n",
    "ax1.plot(vecs[0,:], vecs[1,:], 'r.')\n",
    "ax1.set_xlabel('$x$', fontsize = 16)\n",
    "ax1.set_ylabel('$y$', fontsize = 16)\n",
    "ax1.set_title('$\\mathbb{R}^n$', fontsize = 16)\n",
    "# add a blue star for our solution\n",
    "ax1.plot(37./11,1./11, 'b*',markersize=20)\n",
    "\n",
    "ax2 = fig.add_subplot(1, 2, 2, projection='3d')\n",
    "ax2.plot(Avecs[0,:], Avecs[1,:], Avecs[2,:], 'r.')\n",
    "ax2.set_xlabel('$x$', fontsize = 16)\n",
    "ax2.set_ylabel('$y$', fontsize = 16)\n",
    "ax2.set_zlabel('$z$', fontsize = 16)\n",
    "ax2.set_title('$\\mathbb{R}^m$', fontsize = 16)\n",
    "ax2.set_xlim3d(-15, 15)\n",
    "ax2.set_ylim3d(-15, 15)\n",
    "ax2.set_zlim3d(-15, 15)\n",
    "# add a blue star for the RHS vector\n",
    "ax2.plot([7],[3],[-11],'b*',markersize=20)\n",
    "\n",
    "# rotate to try and get a better view - with a different plotting backend you could rotate with mouse\n",
    "# you could edit this to try and get a better idea of the 3D view\n",
    "ax2.view_init(30, 60)\n",
    "\n",
    "# add the column vectors\n",
    "vec1 = A[:,0]; vec2 = A[:,1]\n",
    "ax2.quiver(0, 0, 0, vec1[0], vec1[1], vec1[2], length =3, linewidths=3, color='black', arrow_length_ratio=0.6, label='column 1 direction')\n",
    "ax2.quiver(0, 0, 0, vec2[0], vec2[1], vec2[2], length =3, linewidths=3, color='black', arrow_length_ratio=0.2, label='column 2 direction')\n",
    "\n",
    "# print out A@b\n",
    "print(A@np.array([37./11,1./11]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The blue star in the left domain given by $ x = 37/11, y = 1/11$ does indeed map to our RHS vector shown as the blue star in the right figure.\n",
    "\n",
    "A homework exercise asks you to use row operations to solve this problem, and to consider when this problem does not have an exact solution.\n",
    "\n",
    "<br><br>\n",
    "\n",
    "Now an example with a different sitaution: the third equation has been chosen randomly, but ensuring that it isn't a linear combination of the other two:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x138a662f788>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0,5,100)\n",
    "\n",
    "y1 = -(2./3.)*x + (7./3.)\n",
    "y2 = (1./4.)*x - (3./4.)\n",
    "y3 = -(1./10.)*x - (1./10.)\n",
    "\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.grid(True)\n",
    "\n",
    "ax1.plot(x,y1,'b', label='$2x+3y=7$')\n",
    "ax1.plot(x,y2,'r', label='$x-4y=3$')\n",
    "ax1.plot(x,y3,'g', label='$x+10y=-1$')\n",
    "\n",
    "ax1.legend(loc='best', fontsize=14)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our system of simultaneous equations is now\n",
    "$$\n",
    "\\begin{align*}\n",
    "  2x + 3y &= 7 \\\\[5pt]\n",
    "   x - 4y &= 3 \\\\[5pt]\n",
    "   x + 10y & = -1\n",
    "\\end{align*}\n",
    "   \\quad \\iff \\quad\n",
    "  \\begin{pmatrix}\n",
    "    2 & 3 \\\\\n",
    "    1 & -4  \\\\\n",
    "    1 & 10 \n",
    "  \\end{pmatrix}\n",
    "  \\begin{pmatrix}\n",
    "    x \\\\\n",
    "    y \n",
    "  \\end{pmatrix}=\n",
    "  \\begin{pmatrix}\n",
    "    7 \\\\\n",
    "    3 \\\\\n",
    "    -1\n",
    "  \\end{pmatrix}   \n",
    "$$\n",
    "\n",
    "We rearranged each to the following in order to easily draw three straight lines plotted above:\n",
    "\n",
    "\\begin{align*}\n",
    "  y &= -\\frac{2}{3}x + \\frac{7}{3} \\\\[5pt]\n",
    "  y &= \\frac{1}{4}x - \\frac{3}{4} \\\\[5pt]\n",
    "  y &= -\\frac{1}{10}x - \\frac{1}{10}\n",
    "\\end{align*}\n",
    "\n",
    "Or plotted in terms of the transformation of points/vectors:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<mpl_toolkits.mplot3d.art3d.Line3DCollection at 0x138a7ee1888>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# let's plot the transformation\n",
    "\n",
    "A = np.array([[2, 3], [1, -4], [1, 10]])\n",
    "\n",
    "x = np.linspace(-5, 5, 20)\n",
    "y = np.linspace(-5, 5, 20)\n",
    "# this creates a mesh of points in 2D\n",
    "xx, yy = np.meshgrid(x, y)\n",
    "# convert to row vectors\n",
    "xxx = np.reshape(xx,(1,np.size(xx)))\n",
    "yyy = np.reshape(yy,(1,np.size(yy)))\n",
    "# convert to a 2 x N matrix of vectors/points\n",
    "vecs = np.vstack((xxx,yyy))\n",
    "\n",
    "# transform these points\n",
    "Avecs = A@vecs\n",
    "\n",
    "fig = plt.figure(figsize=(12, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(1, 2, 1)\n",
    "ax1.plot(vecs[0,:], vecs[1,:], 'r.')\n",
    "ax1.set_xlabel('$x$', fontsize = 16)\n",
    "ax1.set_ylabel('$y$', fontsize = 16)\n",
    "ax1.set_title('$\\mathbb{R}^n$', fontsize = 16)\n",
    "\n",
    "ax2 = fig.add_subplot(1, 2, 2, projection='3d')\n",
    "ax2.plot(Avecs[0,:], Avecs[1,:], Avecs[2,:], 'r.')\n",
    "ax2.set_xlabel('$x$', fontsize = 16)\n",
    "ax2.set_ylabel('$y$', fontsize = 16)\n",
    "ax2.set_zlabel('$z$', fontsize = 16)\n",
    "ax2.set_title('$\\mathbb{R}^m$', fontsize = 16)\n",
    "ax2.set_xlim3d(-15, 15)\n",
    "ax2.set_ylim3d(-15, 15)\n",
    "ax2.set_zlim3d(-15, 15)\n",
    "# add a blue star for the RHS vector\n",
    "ax2.plot([7],[3],[-1],'b*',markersize=20)\n",
    "\n",
    "# rotate to try and get a better view - with a different plotting backend you could rotate with mouse\n",
    "# you could edit this to try and get a better idea of the 3D view\n",
    "ax2.view_init(30, 30)\n",
    "\n",
    "# add the column vectors\n",
    "vec1 = A[:,0]; vec2 = A[:,1]\n",
    "ax2.quiver(0, 0, 0, vec1[0], vec1[1], vec1[2], length =3, linewidths=3, color='black', arrow_length_ratio=0.6, label='column 1 direction')\n",
    "ax2.quiver(0, 0, 0, vec2[0], vec2[1], vec2[2], length =3, linewidths=3, color='black', arrow_length_ratio=0.2, label='column 2 direction')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What would we do in this case if we needed a solution that in some sense satisfied our equations?\n",
    "\n",
    "<br><br><br>\n",
    "\n",
    "We can use the ***least squares solution*** we saw in the previous lecture.\n",
    "\n",
    "<br>\n",
    "\n",
    "Recall that for a given linear problem $A\\boldsymbol{x} = \\boldsymbol{b}$, if we can't solve it exactly then we can instead ask that it's \"nearly\" solved, i.e. we can seek to find $\\boldsymbol{x}$ such that the residual $A\\boldsymbol{x} - \\boldsymbol{b}$ is small. If we measure smallness using the two norm, i.e. minimse $\\|A\\boldsymbol{x} -\\boldsymbol{b}\\|_2$\n",
    "(or equivalently its square if we wish to avoid the square root that appears in the norm), then the solution to this problem is the ***least squares solution***.\n",
    "\n",
    "<br>\n",
    "\n",
    "Recall that we introduced a way to find this solution when we consiered the problem in the form of fitting a line to a cloud of data points in L1 - it involved multiplying through by the transpose of the LHS matrix, i.e. here solving the problem\n",
    "\n",
    "$$A^TA\\boldsymbol{x} = A^T\\boldsymbol{b}$$\n",
    "\n",
    "What is the dimension of $A^TA$ and when do we expect this latest problem to have a unique solution, i.e. when do we expect to be able to find the least squares solution via this route? NB. when we can't solve this equation it doesn't mean the least squares solution doesn't exist, it just means that we need to use a different method to find it.\n",
    "\n",
    "<br>\n",
    "Let's use this approach and see where the solution lies."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x138a8114b48>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = np.array([[2, 3], [1, -4], [1, 10]])\n",
    "\n",
    "# Form the matrix A.T @ A\n",
    "ATA = A.T @ A  \n",
    "\n",
    "# Form the RHS vector:\n",
    "rhs = A.T @ np.array([7,3,-1])\n",
    "\n",
    "# solve the system\n",
    "ls_sol = sl.solve(ATA, rhs)\n",
    "\n",
    "# plot this solution to see where it lies\n",
    "\n",
    "x = np.linspace(0,5,100)\n",
    "\n",
    "y1 = -(2./3.)*x + (7./3.)\n",
    "y2 = (1./4.)*x - (3./4.)\n",
    "y3 = -(1./10.)*x - (1./10.)\n",
    "\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.grid(True)\n",
    "\n",
    "ax1.plot(x,y1,'b', label='$2x+3y=7$')\n",
    "ax1.plot(x,y2,'r', label='$x-4y=3$')\n",
    "ax1.plot(x,y3,'g', label='$x+10y=-1$')\n",
    "ax1.plot(ls_sol[0], ls_sol[1], 'ko', label='LS solution')\n",
    "\n",
    "ax1.legend(loc='best', fontsize=14)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So visually we can see that it is indeed a solution that is doing its best to be on all three lines.\n",
    "\n",
    "\n",
    "If we didn't change the LHS matrix, but only changed the RHS vector, how could we come up with a version of this problem that had a solution that satisfied the system exactly?  Think about the range of the matrix $A$ - see homework."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So the least squares solution to the $3\\times 2$ problem is in some sense an average of the three solutions to the problems we get if we consider 2 out of the 3 constraints/equations separately (these would just be the intersections of each pair of lines of course - as part of a homework exercise these are computed).\n",
    "\n",
    "Said another way, the black dot is trying to satisfy all constraints, i.e. to lie on all three lines, in a least square sense it's satisfying all three the best it can - i.e. if we perturb the point we should find that $\\|Ax-b\\|_2$ grows.\n",
    "\n",
    "See homework exercise where you are asked to do this.\n",
    "\n",
    "<br>\n",
    "\n",
    "We can plot the transformation\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<mpl_toolkits.mplot3d.art3d.Line3DCollection at 0x138a8147548>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# let's plot the transformation\n",
    "\n",
    "A = np.array([[2, 3], [1, -4], [1, 10]])\n",
    "\n",
    "x = np.linspace(-5, 5, 20)\n",
    "y = np.linspace(-5, 5, 20)\n",
    "# this creates a mesh of points in 2D\n",
    "xx, yy = np.meshgrid(x, y)\n",
    "# convert to row vectors\n",
    "xxx = np.reshape(xx,(1,np.size(xx)))\n",
    "yyy = np.reshape(yy,(1,np.size(yy)))\n",
    "# convert to a 2 x N matrix of vectors/points\n",
    "vecs = np.vstack((xxx,yyy))\n",
    "\n",
    "# transform these points\n",
    "Avecs = A@vecs\n",
    "\n",
    "# LS solution\n",
    "ATA = A.T @ A\n",
    "b = np.array([7,3,-1])\n",
    "rhs = A.T @ b\n",
    "ls_sol = sl.solve(ATA, rhs)\n",
    "\n",
    "# plot\n",
    "fig = plt.figure(figsize=(12, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(1, 2, 1)\n",
    "ax1.plot(vecs[0,:], vecs[1,:], 'r.')\n",
    "ax1.set_xlabel('$x$', fontsize = 16)\n",
    "ax1.set_ylabel('$y$', fontsize = 16)\n",
    "ax1.set_title('$\\mathbb{R}^n$', fontsize = 16)\n",
    "# add a black star for our LS solution\n",
    "ax1.plot(ls_sol[0],ls_sol[1], 'k*',markersize=20)\n",
    "\n",
    "ax2 = fig.add_subplot(1, 2, 2, projection='3d')\n",
    "ax2.plot(Avecs[0,:], Avecs[1,:], Avecs[2,:], 'r.')\n",
    "ax2.set_xlabel('$x$', fontsize = 16)\n",
    "ax2.set_ylabel('$y$', fontsize = 16)\n",
    "ax2.set_zlabel('$z$', fontsize = 16)\n",
    "ax2.set_title('$\\mathbb{R}^m$', fontsize = 16)\n",
    "ax2.set_xlim3d(-15, 15)\n",
    "ax2.set_ylim3d(-15, 15)\n",
    "ax2.set_zlim3d(-15, 15)\n",
    "# black star for A@ls_sol\n",
    "Als_sol = A@ls_sol\n",
    "ax2.plot([Als_sol[0]],[Als_sol[1]],[Als_sol[2]],'k*',markersize=20)\n",
    "# blue star for our RHS vector\n",
    "ax2.plot([b[0]],[b[1]],[b[2]],'b*',markersize=20)\n",
    "\n",
    "# rotate the view - you could edit this to try and get a better idea of the 3D view\n",
    "ax2.view_init(140, 80)\n",
    "\n",
    "# add the column vectors\n",
    "vec1 = A[:,0]; vec2 = A[:,1]\n",
    "ax2.quiver(0, 0, 0, vec1[0], vec1[1], vec1[2], length =3, linewidths=3, color='black', arrow_length_ratio=0.6, label='column 1 direction')\n",
    "ax2.quiver(0, 0, 0, vec2[0], vec2[1], vec2[2], length =3, linewidths=3, color='black', arrow_length_ratio=0.2, label='column 2 direction')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.000000e+00 -3.552714e-15]\n"
     ]
    }
   ],
   "source": [
    "# Q. What am I establishing/checking here????\n",
    "print( A.T @ (A@ls_sol - b))\n",
    "# A. That the columns of the matrix A are orthogonal to ... what??"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's consider another case where one equation is repeated but with a different RHS value (i.e. equivalently the corresponding line has the same slope but different intercept). This is just another way we can arrive at an inconsistent set of equations/constraints. We can still try to find a useful solution. Let's apply the same solution and plotting procedure and see if the solution and plot we get makes sense."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x138a8261b88>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# another example\n",
    "\n",
    "A = np.array([[2, 3], [1, -4], [2, 3]])\n",
    "\n",
    "# Form the matrix A.T @ A\n",
    "ATA = A.T @ A  \n",
    "\n",
    "# Form the RHS vector:\n",
    "rhs = A.T @ np.array([7,3,1])\n",
    "\n",
    "# solve the system\n",
    "ls_sol = sl.solve(ATA, rhs)\n",
    "\n",
    "# plot this solution to see where it lies\n",
    "\n",
    "x = np.linspace(0,5,100)\n",
    "\n",
    "y1 = -(2./3.)*x + (7./3.)\n",
    "y2 = (1./4.)*x - (3./4.)\n",
    "#y3 = -(1./10.)*x - (1./10.)\n",
    "y3 = -(2./3.)*x + (1./3.)\n",
    "\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.grid(True)\n",
    "\n",
    "ax1.plot(x,y1,'b', label='$2x+3y=7$')\n",
    "ax1.plot(x,y2,'r', label='$x-4y=3$')\n",
    "ax1.plot(x,y3,'g', label='$x+10y=-1$')\n",
    "ax1.plot(ls_sol[0], ls_sol[1], 'ko', label='LS solution')\n",
    "\n",
    "ax1.legend(loc='best', fontsize=14)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A simple under-determined case\n",
    "\n",
    "This is the opposite situation where we have now less equations/constraints than unknowns. \n",
    "\n",
    "We might therefore also call it an under-constrained problem and if we can find one exact solution, then there will actually be  infinitely many solutions.\n",
    "\n",
    "The simplest case would be one equation, two unknowns. We already know that assuming we restrict ourselves to linear problems, this one equation featuring two unknowns just corresponds to a straight line, e.g. let's suppose we for some reason had the first equation from the above problem (we were only able to collect 1 rather than 3 pieces of data say)\n",
    "\n",
    "$$2x + 3y = 7\n",
    "   \\quad \\iff \\quad\n",
    "  \\begin{pmatrix}\n",
    "    2 & 3 \n",
    "  \\end{pmatrix}\n",
    "  \\begin{pmatrix}\n",
    "    x \\\\\n",
    "    y \n",
    "  \\end{pmatrix}=\n",
    "  \\begin{pmatrix}\n",
    "    7 \n",
    "  \\end{pmatrix}   \n",
    "$$\n",
    "\n",
    "We know that any $(x,y)$ pair that satisfies this is a valid solution given the information we have.\n",
    "\n",
    "As for the previous case, can we come up with a solution method that gives us a sensible single unique solution?\n",
    "\n",
    "Yes, it's called the ***minimum norm solution*** and is given by\n",
    "\n",
    "$$\\boldsymbol{x} = A^T(AA^T)^{-1}\\boldsymbol{b}$$\n",
    "\n",
    "Think again about the dimensions of each of the variables and terms appearing in this - does it make dimensional sense?\n",
    "\n",
    "\n",
    "Now in this simple case the $AA^T$ matrix we need to take the inverse of is actually just a scalar, so this is trivial to evaluate here, but this formula extends to arbitrary sized under-determined problems."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x138a6616908>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# come up with an example with indpt equations\n",
    "A = np.array([[2, 3]])\n",
    "b = np.array([7])\n",
    "\n",
    "# construct the right inverse:\n",
    "\n",
    "A_ri = A.T @ sl.inv(A@A.T)\n",
    "\n",
    "x_m = A_ri @ b\n",
    "\n",
    "# check that this is a solution:   Ax_m = b?\n",
    "print(np.allclose(b, A@x_m))\n",
    "\n",
    "# and plot it\n",
    "\n",
    "x = np.linspace(-1,3,100)\n",
    "\n",
    "y1 = -(2./3.)*x + (7./3.)\n",
    "\n",
    "fig = plt.figure(figsize=(5, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(111)\n",
    "\n",
    "ax1.set_xlabel(\"$x$\", fontsize=14)\n",
    "ax1.set_ylabel(\"$y$\", fontsize=14)\n",
    "ax1.grid(True)\n",
    "\n",
    "ax1.plot(x,y1,'b', label='$2x+3y=7$')\n",
    "\n",
    "\n",
    "ax1.plot(x_m[0], x_m[1], 'ko', label='min norm solution')\n",
    "\n",
    "ax1.legend(loc='best', fontsize=14)\n",
    "\n",
    "# set axis equal so we can get a good idea of a 90 degree angle\n",
    "ax1.axis('equal')\n",
    "ax1.plot([0,x_m[0]], [0,x_m[1]], 'k--')\n",
    "ax1.plot(0,0,'ro')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The black dot is the solution this method obtains. The red dot is the line between this and the **origin** (i.e. the point $(0,0)$ with norm zero) given by the red dot, and I have drawn the black dashed line just to emphasise that it's at right angles (is orthogonal) to the blue line, hence it's the closest point in the blue line to the origin.\n",
    "\n",
    "<br>\n",
    "\n",
    "[Tip: it you want to make a point about angles visually, it helps to use the same spacing in the $x$ and $y$ direction in plots - with Matplotlib: `ax1.axis('equal')`. Rerun without this line to see the point I'm making here].\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "This procedure therefore finds the solution (out of the infinitely many that lie along the blue line) that is closest to the origin - hence the phrase ***minimum norm solution*** - out of all the possible solution vectors $\\boldsymbol{x}$ it's the one with the smallest value of $\\|\\boldsymbol{x}\\|_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### An under-determined problem with no solutions, and with infinitely many solutions\n",
    "\n",
    "We still need existence of solutions in the under-determined case (if we do have existence then we automatically have non-uniqueness).\n",
    "\n",
    "We have no solutions when the equations are inconsistent. \n",
    "\n",
    "[We can't demonstrate this in the one equation / two+ unknown case of course, we need to move to at least two equations for them to be inconsistent.]\n",
    "\n",
    "For example, the problem with two equations and three unknowns:\n",
    "\n",
    "$$ \n",
    "\\begin{align*}\n",
    "x + y + z = 0 \\\\[5pt]\n",
    "x + y + z = 1\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "has no solutions, \n",
    "\n",
    "\n",
    "<br><br>\n",
    "\n",
    "whereas\n",
    "\n",
    "$$ \n",
    "\\begin{align*}\n",
    "x + y + z = 0 \\\\[5pt]\n",
    "x + y - z = 1\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "has infinitely many.\n",
    "\n",
    "<br>\n",
    "\n",
    "Let's confirm using the method just introduced.\n",
    "\n",
    "\n",
    "<br><br>\n",
    "\n",
    "First the case with inconsistent equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "det(A@A.T) =  0.0\n",
      "\n",
      "Marix A@A.T is singular:  True\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[1, 1, 1], [1, 1, 1]])\n",
    "b = np.array([0, 1])\n",
    "\n",
    "# construct the right inverse:\n",
    "#A_ri = A.T @ sl.inv(A@A.T)\n",
    "\n",
    "# actually before we do this let's check that the right inverse exists, \n",
    "# i.e. can we actually find the inverse of A@A.T\n",
    "print('det(A@A.T) = ', sl.det(A@A.T))\n",
    "print('\\nMarix A@A.T is singular: ', np.isclose(sl.det(A@A.T), 0.))\n",
    "\n",
    "# so let's not bother using the right inverse if it doesn't exist\n",
    "#x_m = A_ri @ b\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the case with consistent equations, which will have infinitely many solutions\n",
    "\n",
    "Can we work out the minimal norm solution ourselves?\n",
    "\n",
    "Our system takes the form\n",
    "\n",
    "$$ \n",
    "\\begin{align*}\n",
    "x + y + z = 0 \\\\[5pt]\n",
    "x + y - z = 1\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "Let's progress as we normally would with simultaneous equations and try to cancel variables. \n",
    "\n",
    "Notice that we can immediately cancel two variables if we subtract the second from the first:\n",
    "\n",
    "$$ 2z = -1 \\Rightarrow z = -\\frac{1}{2}$$\n",
    "\n",
    "Substituting this value for $z$ into the other two equations, we find that we are actually left with a repeated equation\n",
    "\n",
    "$$ \n",
    "\\begin{align*}\n",
    "x + y = \\frac{1}{2} \\\\[5pt]\n",
    "x + y = \\frac{1}{2}\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "these are consistent as both the LHS's and the RHS's are consistent, i.e. this is effectively one equation for two unknowns, and so has infinitely many solutions that can be written\n",
    "\n",
    "$$ \n",
    "\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "\\alpha \\\\\n",
    "\\frac{1}{2} - \\alpha \\\\\n",
    "-\\frac{1}{2}\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "for any $\\alpha \\in \\mathbb{R}$. \n",
    "\n",
    "Out if this infinitely many solutions, the minimum norm solution is the one that minimises the solution vector's two norm, i.e. which minimses the quantity\n",
    "\n",
    "$$\\alpha^2 + \\left(\\frac{1}{2} - \\alpha\\right)^2 + \\left(-\\frac{1}{2}\\right)^2 = \n",
    "\\alpha^2 + \\frac{1}{4} - \\alpha  + \\alpha^2 + \\frac{1}{4}\n",
    "= 2\\alpha^2 - \\alpha + \\frac{1}{2}$$\n",
    "\n",
    "Taking derivatives and setting to zero ($4\\alpha - 1 = 0$) , this is minimised when $\\alpha = \\frac{1}{4}$, and then our solution is\n",
    "\n",
    "$$ \n",
    "\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "\\frac{1}{4} \\\\[5pt]\n",
    "\\frac{1}{4} \\\\[5pt]\n",
    "-\\frac{1}{2}\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "Do we get this solution when we use the formula from above\n",
    "\n",
    "$$\\boldsymbol{x} = A^T(AA^T)^{-1}\\boldsymbol{b}$$\n",
    "\n",
    "?  \n",
    "\n",
    "Let's check:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.25  0.25 -0.5 ]\n",
      "True\n"
     ]
    }
   ],
   "source": [
    "# our example with indpt equations\n",
    "A = np.array([[1, 1, 1], [1, 1, -1]])\n",
    "b = np.array([0, 1])\n",
    "\n",
    "# construct the right inverse:\n",
    "\n",
    "A_ri = A.T @ sl.inv(A@A.T)\n",
    "\n",
    "x_m = A_ri @ b\n",
    "\n",
    "# print the solution\n",
    "print(x_m)\n",
    "\n",
    "# check that this is a solution:   Ax_m = b?\n",
    "print(np.allclose(b, A@x_m))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.quiver.Quiver at 0x138a804d088>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x360 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# let's plot the transformation\n",
    "\n",
    "A = np.array([[1, 1, 1], [1, 1, -1]])\n",
    "\n",
    "\n",
    "x = np.linspace(-1, 1, 10)\n",
    "y = np.linspace(-1, 1, 10)\n",
    "z = np.linspace(-1, 1, 10)\n",
    "# this creates a mesh of points in 2D\n",
    "xx, yy, zz = np.meshgrid(x, y, z)\n",
    "# convert to row vectors\n",
    "xxx = np.reshape(xx,(1,np.size(xx)))\n",
    "yyy = np.reshape(yy,(1,np.size(yy)))\n",
    "zzz = np.reshape(zz,(1,np.size(zz)))\n",
    "# convert to a 2 x N matrix of vectors/points\n",
    "vecs = np.vstack((xxx,yyy,zzz))\n",
    "\n",
    "# transform these points\n",
    "Avecs = A@vecs\n",
    "\n",
    "# the family of possible solutions, just a selection based on a restricted range of alpha values\n",
    "alpha = np.linspace(-2, 2, 10)\n",
    "x_sols = np.vstack((alpha, 1/2 - alpha, -1/2*np.ones_like(alpha)))\n",
    "# and check where they map to\n",
    "Ax_sols = A@x_sols\n",
    "\n",
    "# plot\n",
    "fig = plt.figure(figsize=(12, 5))\n",
    "\n",
    "ax1 = fig.add_subplot(1, 2, 1, projection='3d')\n",
    "ax1.plot(vecs[0,:], vecs[1,:], vecs[2,:], 'r.')\n",
    "ax1.set_xlabel('$x$', fontsize = 16)\n",
    "ax1.set_ylabel('$y$', fontsize = 16)\n",
    "ax1.set_zlabel('$z$', fontsize = 16)\n",
    "ax1.set_title('$\\mathbb{R}^n$', fontsize = 16)\n",
    "ax1.set_xlim3d(-1, 1)\n",
    "ax1.set_ylim3d(-1, 1)\n",
    "ax1.set_zlim3d(-1, 1)\n",
    "ax1.plot(x_sols[0,:], x_sols[1,:], x_sols[2,:], 'b*',markersize=20)\n",
    "                        \n",
    "\n",
    "ax2 = fig.add_subplot(1, 2, 2)\n",
    "ax2.plot(Avecs[0,:], Avecs[1,:], 'r.')\n",
    "ax2.set_xlabel('$x$', fontsize = 16)\n",
    "ax2.set_ylabel('$y$', fontsize = 16)\n",
    "ax2.set_title('$\\mathbb{R}^m$', fontsize = 16)\n",
    "ax2.plot(Ax_sols[0,:], Ax_sols[1,:], 'b*',markersize=20)\n",
    "\n",
    "# add the column vectors\n",
    "vec1 = A[:,0]; vec2 = A[:,1]; vec3 = A[:,2]\n",
    "ax2.quiver(0, 0, vec1[0], vec1[1] , angles='xy', units='xy', scale=1, color='black', width=.1,  label='vec1')\n",
    "ax2.quiver(0, 0, vec2[0], vec2[1] , angles='xy', units='xy', scale=1, color='black', width=.1,  label='vec2')\n",
    "ax2.quiver(0, 0, vec3[0], vec3[1] , angles='xy', units='xy', scale=1, color='black', width=.1,  label='vec3')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We only see two vectors on the right, even though we've plotted all three columns of $A$, as two of the columns are repeated, these are the first two columns and so hopefully the family of solutions we found\n",
    "\n",
    "$$ \n",
    "\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "\\alpha \\\\\n",
    "\\frac{1}{2} - \\alpha \\\\\n",
    "-\\frac{1}{2}\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "makes sense in this light, the arbitrary $\\alpha$ in $x$ is completely cancelled out by the $-\\alpha$ that appears in $y$ (after we pre-multiply by $A$ in which the first two colums are equal)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A summary of what we've establised so far about solution methods\n",
    "\n",
    "We have seen that in the ***over-determined case***, where we can't in general satisfy all equations/constraints, that the ***least squares solution*** gave us one useful solution, and could be found by solving the square linear system:\n",
    "\n",
    "$$A^TA\\boldsymbol{x} = A^T\\boldsymbol{b}$$\n",
    "\n",
    "or assuming the inverse can be taken:\n",
    "\n",
    "$$\\boldsymbol{x} = (A^TA)^{-1}A^T\\boldsymbol{b}$$\n",
    "\n",
    "\n",
    "<br><br>\n",
    "\n",
    "\n",
    "While in the ***under-determined case***, assuming there is no inconsistency between equations, we have infinitely many solutions. The solution being closest to the origin, i.e. the ***minimum-norm solution***, could be obtained using the formula\n",
    "\n",
    "$$\\boldsymbol{x} = A^T(AA^T)^{-1}\\boldsymbol{b},$$\n",
    "\n",
    "in this case assuming that $AA^T$ is invertible.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Over-, under-, equi- and mixed-determined problems\n",
    "\n",
    "## Introduction\n",
    "\n",
    "Consider the linear inversion problem represented by the equation \n",
    "\n",
    "$$G\\, \\boldsymbol{m} = \\boldsymbol{d} \\quad \\iff \\quad A\\, \\boldsymbol{x} = \\boldsymbol{b} $$\n",
    "\n",
    "where $G$ is an $m \\times n$ matrix, we can formally identify several possible circumstances:  \n",
    "\n",
    "\n",
    "\n",
    "### The equi-determined case\n",
    "\n",
    "<img src=\"./latex/matrix-square.png\" width=400x>\n",
    "\n",
    "There  are  exactly  as  many  data  points  as  model  parameters.  \n",
    "\n",
    "The  matrix  $G$  will  be  square  so  that  $m  =  n$. \n",
    "\n",
    "There  will  be  as  many equations as unknowns.  \n",
    "\n",
    "<br>\n",
    "\n",
    "If these equations are both *independent*\n",
    "\n",
    "(i.e. the \"lines\" we saw previously in $2\\times 2$ cases corresponding to each equation/constraint are not parallel) \n",
    "\n",
    "are and *consistent*\n",
    "\n",
    "(which means that the lines cross at least once; inconsistency is where we can derive a contradiction from the equations, e.g. using row operations to arrive at the equation $0=1$, i.e. the rank of the augmented matrix is greater than the rank of the coefficient matrix - see next cell), \n",
    "\n",
    "then the problem is called ***equi-determined***.  \n",
    "\n",
    "There will be one and only one solution that will fit the data (the RHS vector) exactly.  \n",
    "\n",
    "<br><br>\n",
    "\n",
    "For further discussion of this see <https://en.wikipedia.org/wiki/System_of_linear_equations#Properties>\n",
    "\n",
    "also ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aside - the Rouché–Capelli theorem\n",
    "\n",
    "The [Rouché–Capelli theorem](https://en.wikipedia.org/wiki/Rouch%C3%A9%E2%80%93Capelli_theorem) tells us that  \n",
    "\n",
    "\"any system of equations ... is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank; hence in such a case there are an infinitude of solutions.\"\n",
    "\n",
    "Above quoted from <https://en.wikipedia.org/wiki/Overdetermined_system#Non-homogeneous_case>\n",
    "\n",
    "\n",
    "<br><br>\n",
    "\n",
    "We saw a simple example of an inconsistent system earlier:\n",
    "\n",
    "$$ \n",
    "\\begin{align*}\n",
    "x + y + z = 0 \\\\[5pt]\n",
    "x + y + z = 1\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "if you form the augmented matrix here and perform row operations, the rank of the augmented matrix will clearly be larger than the rank of $A$ alone. Just drop the $z$ from these equations and we have an inconsistent square system."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The over-determined case\n",
    "\n",
    "<img src=\"./latex/matrix-tall.png\" width=300x>\n",
    "\n",
    "\n",
    "In this case there  are  more  independent  data  points  than  there  are  model parameters.\n",
    "\n",
    "The matrix $G$ will be a \"tall\" matrix with more rows than columns so that $m > n$.  \n",
    "\n",
    "Equivalently, there are more equations (or constraints) than unknowns. \n",
    "\n",
    "If $G$ is full rank (recall that this means that $\\text{rank}(G) = \\min(m,n)= n$), then the inversion problem is called ***purely over-determined***.  \n",
    "\n",
    "\n",
    "Typically there will be no exact solution to an over-determined problem.   \n",
    "\n",
    "In the purely over-determined case we can use our formula above for the least squares solution.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### The under-determined case\n",
    "\n",
    "\n",
    "<img src=\"./latex/matrix-fat.png\" width=400x>\n",
    "\n",
    "\n",
    "In this case there  are  fewer data points than model parameters.  \n",
    "\n",
    "The matrix $G$ will be a \"fat\" matrix with fewer rows than columns so that $m < n$.  \n",
    "\n",
    "Equivalently, there are less equations (or constraints) than unknowns. \n",
    "\n",
    "If  $G$  is  full  rank (recall that this means that $\\text{rank}(G) = \\min(m,n) = m$), then the problem is called ***purely under-determined***. \n",
    "\n",
    "There will be infinitely many solutions that are able to fit the data exactly.\n",
    "\n",
    "<br>\n",
    "\n",
    "If an under-determined case is **not** purely under-determined (i.e. $\\text{rank}(G) < m$), then this means that we can't find solutions that exactly fit the data.\n",
    "\n",
    "<br>\n",
    "\n",
    "We've seen simple examples of purely under-determined problems previously.\n",
    "\n",
    "The following is an example that is **not** purely under-determined, as it is indeed under-determined but has no exact solution(s) as the equations are inconsistent:\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "1 & 2 & 3 \\\\\n",
    "2 & 4 & 6\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "m_1\\\\\n",
    "m_2\\\\\n",
    "m_3\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "1\\\\\n",
    "1\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### The mixed-determined case\n",
    "\n",
    "\n",
    "Many real problems will actually be partly over-determined and partly under-determined.  \n",
    "\n",
    "These problems  are  called  ***mixed-determined***. \n",
    "\n",
    "Problems  in  which  some  but  not  all  of  the parameters  are  equi-determined  are  also  mixed  determined. \n",
    "\n",
    "In  mixed-determined problems,  some of the model parameters (i.e. some components of the unknown $\\boldsymbol{x}$)  will  typically  be  described  by  more independent equations than there are parameters and there will be no model that can satisfy this set of equations, \n",
    "\n",
    "while some other parameters will be described by fewer independent equations than there are parameters and there may be be infinitely many models that can satisfy this second set of equations.  \n",
    "\n",
    "A mixed-determined problem can occur with any shaped matrix $G$.  \n",
    "\n",
    "There will typically be no exact solution to this problem.  In addition, there will be infinitely many solutions that can fit the data equally accurately. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix rank in the non-square case \n",
    "\n",
    "When $A$ is of size $m\\times n$ we can think of $A$ as mapping vectors in $\\mathbb{R}^n$ into $\\mathbb{R}^m$ - matrix-multiplication can be interpreted as moving in a linear combination of directions given by the columns of the matrix in the space $\\mathbb{R}^m$. \n",
    "\n",
    "\n",
    "In the non-square case $A$ is said to have *full rank* if $\\text{rank}(A)=\\min(m,n)$ and is said to be *rank deficient* if $\\text{rank}(A)<\\min(m,n)$.  \n",
    "\n",
    "\n",
    "Note that as the column and row rank are also equal for non-square matrices, i.e. the number of linearly independent rows is the same as the number of linearly independent columns, then of course the maximum the rank of the matrix can ever be is $\\min(m,n)$\n",
    " \n",
    "<br>\n",
    "\n",
    "Assuming $m\\ne n$ then there are two possibilities:\n",
    "\n",
    "\n",
    "1. In the \"tall\" [over-determined](https://en.wikipedia.org/wiki/Overdetermined_system) case ($m>n$), the range (or column space) of $A$ doesn't (cannot) cover all of the space that our data $\\boldsymbol{b}$ lives in, so we cannot find exact solutions for all data. But if $\\boldsymbol{b}$ does live in the range then we can find a solution (it may or may not be unique).\n",
    "\n",
    "\n",
    "2. In the \"fat\" [under-determined](https://en.wikipedia.org/wiki/Underdetermined_system) case ($m<n$) it's the opposite, if the rank of $A$ is larger than the dimension of the data space (as will be the case if the matrix is full rank) then there will always be multiple linear combinations of the columns that get us to any point in the vector space containing the data, i.e. we will always have a non-unique solution for all RHS $\\boldsymbol{b}$ vectors. In the above $2\\times 3$ case the rank of $A$ is only 1 which is less than the dimension of the data space which here is 2 - this reiterates that we don't necessarily expect solutions in this case.\n"
   ]
  },
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   "cell_type": "markdown",
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    "### Example: matrix (outer) products\n",
    "\n",
    "Note that a matrix product always has a rank that is less than or equal to the smallest rank of any of the constituent  matrices.    \n",
    "\n",
    "For example, the outer product $\\boldsymbol{a}\\boldsymbol{b}^T$ of two column  vectors must be rank 1 because vectors are only rank 1. \n",
    "\n",
    "<br>\n",
    "\n",
    "You could check this with an example - see homework.\n",
    "\n",
    "<br>\n",
    "<br>\n",
    "\n",
    "Several important results follow from this observation:\n",
    "\n",
    "\n",
    "- A square $(m = n)$  matrix is full rank if and only if it is non-singular.  \n",
    "\n",
    "\n",
    "- For a tall $(m > n)$ matrix $A$ that is full rank, the matrix formed by $A^TA$ (which has dimension $n \\times n$) will be non-singular.  \n",
    "\n",
    "\n",
    "- For a fat $(m < n)$ matrix $A$ that is full rank, the matrix formed by $AA^T$ (which has dimension $m \\times m$) will be non-singular. \n",
    " \n",
    " \n",
    "- If $A$ is not full rank, then whatever the shape of $A$, both these matrices will be singular.  \n",
    "\n",
    "<br>\n",
    "\n",
    "Note that the first three cases correspond respectively to equi-, purely over-, and purely under- determined inversion problems. The mixed-determined case will fall into category 4."
   ]
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    "Why is this important?\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "- Recall in the under-determined case we used the following expression to give us the *minimum norm solution*\n",
    "\n",
    "$$\\boldsymbol{x} = A^T(AA^T)^{-1}\\boldsymbol{b}$$\n",
    "\n",
    "\n",
    "\n",
    "- In the over-determined case we used the following to give us the *least squares solution*\n",
    "\n",
    "$$\\boldsymbol{x} = (A^TA)^{-1} A^T\\boldsymbol{b}$$\n",
    "\n",
    "\n",
    "<br>\n",
    "\n",
    "Establishing that $A$ has full rank, no matter its shape or size, allows us to use these two explicit formulae as appropriate.\n",
    "\n",
    "<br>\n",
    "\n",
    "[And of course in the square, equi-determined case means we have an inverse matrix.]\n",
    "\n",
    "<br>\n",
    "\n",
    "But we clearly have a problem in finding a solutiion method for problems in case 4, which includes mixed-determined problems."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Some examples\n",
    "\n",
    "### Example 1\n",
    "\n",
    "Consider the following problem:\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "1 & 0 & 0 \\\\\n",
    "1 & 0 & 0 \\\\\n",
    "0 & 2 & 2 \\\\\n",
    "0 & 3 & 3\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "m_1\\\\\n",
    "m_2\\\\\n",
    "m_3\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "2\\\\\n",
    "3\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "For this problem, $m_1$ is over-determined since there is no possible value of $m_1$ that can exactly fit both of the first two equations in this system - the first two equations/constraints are inconsistent. \n",
    "\n",
    "The problem is also under-determined because the last two equations are not independent. In the particular case as written here the final two equations actually boil down to one equation for two unknowns. Thus, there are infinitely many solutions for $m_2$ and $m_3$ that can fit the last two equations exactly.  \n",
    "\n",
    "This problem is therefore clearly ***mixed-determined***.  \n",
    "\n",
    "A homework exercise (next lecture) asks you to come up with a sensible/reasonable \"solution\" to this problem.\n",
    "\n",
    "<br>\n",
    "\n",
    "\n",
    "Most, large, real-world, inversion problems are likely to be mixed-determined.  In practice it can also be very difficult to discover if a large system is mixed determined or not, and so in practical problems we will often proceed by assuming that the problem is mixed-determined, employing an iterative numerical approach.   \n",
    "\n",
    "\n",
    "<br><br>\n",
    "\n",
    "### Example 2\n",
    "\n",
    "The following figure illustrates these four problem types for a simple tomographic experiment. The \"s\" values plotted just indicate that some measurement has been taken, and we want to use this to establish values in the two blocks.\n",
    "\n",
    "\n",
    "<img src=\"./figures/over_under_mixed_determined_example.png\" width=600x>\n",
    "\n",
    "\n",
    "<br><br>\n",
    "\n",
    "- In the first case we have exactly enough information to solve our inversion problem exactly.\n",
    "\n",
    "\n",
    "- In the second case we can only describe the mean of the parameter values, but without additional information there is an infinite family of possible individual parameter values that agree with our data - we have an under-determined problem.\n",
    "\n",
    "\n",
    "- In the third we are given the parameters in each rectangle and additionally the mean - we have an over-determined problem; we may or may not be able to find a solution that satsifies all observations/equations.\n",
    "\n",
    "\n",
    "- In the fourth we are only told the mean (so are under-determined as far as the individual block values, or their difference, are concerned - so these quantities are under-determined) but we are told that mean value three times (so we are over-determined in terms of the mean - an issue if the three data values are different). We could transform this problem into one for two new parameters - the mean and the difference - where we have three equations for the first parameter and none for the second."
   ]
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   "source": [
    "### Example 3 - A tomography inspired mixed-determined problem\n",
    "\n",
    "A relatively simple example that demonstrates a mixed-determined problem that cannot be solved using the methods above is that of estimating the spatial distribution of slowness (the reciprocal of velocity) from measurements of travel times along several straight ray paths through a solid body.  \n",
    "\n",
    "We saw a $4\\times 4$ version of this problem in the cell on tomography in an earlier lecture - this cell is now repeated for completeness.\n",
    "\n",
    "**Straight-ray tomography (repeated from previous lecture)**\n",
    "\n",
    "[\"Tomography is imaging by sections or sectioning, through the use of any kind of penetrating wave\"](https://en.wikipedia.org/wiki/Tomography).\n",
    "\n",
    "Most acoustic and seismic inverse problems are non-linear (since changing the model parameters - the structure of the body being imaged will change the ray path (e.g. due to refraction or reflections) the wave travels along, and hence change the arrival times (the data) in a nonlinear manner).  \n",
    "\n",
    "However, if we assume ray that paths do not vary with the model (which we could justify if we assume that we don't change the $\\boldsymbol{m}$ values very much), and if the data are travel times and the model is composed of [slowness](https://en.wikipedia.org/wiki/Slowness_(seismology)) values, then the problem is linear. \n",
    "\n",
    "Consider the following example\n",
    "\n",
    "<img src=\"./figures/straight_ray_tomog_schematic.png\" width=300x>\n",
    "\n",
    "\n",
    "Suppose that a wall is assembled from an array of bricks as above, and that each brick is composed of a different type of clay.  If the acoustic velocities of the different clays differ, then one might attempt to distinguish the different kinds of bricks by measuring the travel time of sound across the various rows and columns of bricks, in the wall. The data in this problem are $n = 8$ measurements of travel time (based on there being 4 rows and 4 columns) \n",
    "\n",
    "$$ \\boldsymbol{d} = [T_1 , T_2 , T_3 , \\ldots, T_8]^T$$\n",
    "\n",
    "We will assume that each brick is composed of a uniform material and that the travel time of sound  across  each  brick  is  proportional  to  the  width  and  height  of  the  brick  $h$.  The proportionality factor is the brick’s slowness $s_i$, giving 16 model parameters \n",
    "\n",
    "$$ \\boldsymbol{s} = [s_1 , s_2 , s_3 , \\ldots, s_{16}]^T$$\n",
    "\n",
    "where the ordering of the model parameters is according to the numbering scheme in the schematic above.  The  travel  time  of  acoustic  waves  through  the  rows  and  columns  of  a  square  array  of bricks is measured with the acoustic source and receiver placed on the edges of the square.   \n",
    "\n",
    "The set of linear simultaneous equations that we must now solve are: \n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "hs_1+hs_2 + hs_3 + hs_4 &= T_1\\\\\n",
    "hs_5+hs_6 + hs_7 + hs_8 &= T_2\\\\\n",
    "&\\vdots \\\\\n",
    "hs_4+hs_8 + hs_{12} + hs_{16} &= T_8\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "where we order our firing of waves from left to right, starting at the top moving down, and then from top to bottom starting at the left and moving right.\n",
    "\n",
    "These algebraic equations can be arranged as a matrix equation $G\\, \\boldsymbol{m} = \\boldsymbol{d}$ of the form\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "h & h & h & h & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
    "0 & 0 & 0 & 0 & h & h & h & h & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n",
    "\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n",
    "0 & 0 & 0 & h & 0 & 0 & 0 & h & 0 & 0 & 0 & h & 0 & 0 & 0 &h     \\\\\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "s_1\\\\\n",
    "s_2\\\\\n",
    "\\vdots \\\\\n",
    "s_{16}\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "T_1\\\\\n",
    "T_2\\\\\n",
    "\\vdots \\\\\n",
    "T_8\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "This tomography problem is linear because the assumptions we have made were designed to make the travel times depend linearly upon the model parameters.  In addition, slowness rather than velocity is  used for the  model parameters, and slowness  has a linear relationship to  travel time while velocity does not.   \n",
    "\n",
    "X-ray  tomography  used  in  medical  imaging,  and  radar  tomography  used  in  atmospheric physics,  typically  also  assume  ray  paths  that  are  model  independent  and  employ  model parameterisations that are related linearly to the data, and so are able to use linear inversion methods.  In seismic tomography in the interior of the Earth, the independence of the ray path and  the  velocity  model  is  generally  a  poor  assumption  -  in  reality  ray  paths  change significantly when seismic models change, so that linear inversion is not normally adequate for seismic tomography. "
   ]
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    "**A $3\\times 3$ case**\n",
    "\n",
    "In a homework exercise you are asked to consider a $3\\times 3$ case where our equations turn out to be of the form\n",
    "\n",
    "$$\\begin{align*}\n",
    "T_1 &= x_1 + x_2 + x_3\\\\\n",
    "T_2 &= x_4 + x_5 + x_6\\\\\n",
    "& \\vdots \\\\\n",
    "T_6 &= x_3 + x_6 + x_9\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "and hence our matrix system is\n",
    "\n",
    "$$\n",
    "\\begin{pmatrix}\n",
    "1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
    "0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\\\\n",
    "\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots  \\\\\n",
    "0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1\n",
    "\\end{pmatrix}\n",
    "\\begin{pmatrix}\n",
    "x_1\\\\\n",
    "x_2\\\\\n",
    "\\vdots \\\\\n",
    "x_9\n",
    "\\end{pmatrix}\n",
    "=\n",
    "\\begin{pmatrix}\n",
    "T_1\\\\\n",
    "T_2\\\\\n",
    "\\vdots \\\\\n",
    "T_6\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "This problem is taken up in the homework exercises (again, next week)."
   ]
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    "### The issue of imperfect data\n",
    "\n",
    "Let's now consider the issue of imperfect data.\n",
    "\n",
    "If in the previous $3\\times 3$ example all the measurements were perfect and returned the value 6, then clearly the solution with all values 2 is a possible model parameter solution.\n",
    "\n",
    "However, consider the case where the measurements are noisy, let us assume that the following measurements were made for the travel times \n",
    "\n",
    "$$ \\boldsymbol{T} = (6.07, 6.07, 5.77, 5.93, 5.93, 6.03)^T $$\n",
    "\n",
    "Now, even though there are fewer data than model parameters, there is no model that fits the \n",
    "data exactly.  We can see this because it should be the case from $G$ that\n",
    "\n",
    "$$ T_1  + T_2  + T_3  = T_4  + T_5  + T_6$$  \n",
    "\n",
    "whereas for this data\n",
    "\n",
    "$$ T_1 + T_2  + T_3  = 17.91, \\qquad T_4 + T_5  + T_6  = 17.89 $$\n",
    "\n",
    "so that, with these data, there can be no solution. \n",
    "\n",
    "<br><br><br>\n",
    "\n",
    "**How to proceed?**\n",
    "\n",
    "In this problem, $G$ is not square so that $G^{-1}$ does not exist,  \n",
    "\n",
    "in addition both $G^TG$ and $GG^T$ are singular matrices, and hence none of the solution methods that we have used so far will work.  \n",
    "\n",
    "<br>\n",
    "\n",
    "So what can we do?  There are two principal options: \n",
    "\n",
    "\n",
    "1. we can use the ***generalised inverse*** $G^+$, also known as the pseudo-inverse or the [Moore-Penrose inverse](https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse), or\n",
    "\n",
    "\n",
    "2. we can use some form of ***regularisation*** to the model of which ***damped least-squares*** is the most straightforward.   \n",
    "\n",
    "<br>\n",
    "\n",
    "The generalised inverse is  preferable in  small problems, especially  when we would like to analyse the quality of the results carefully, while regularised least-squares and related methods are preferable for large problems when the generalised inverse is prohibitively expensive, or when linearised inversion is being used in order to solve a non-linear problem by iteration.  \n",
    "\n",
    "We will return to this in the next lecture and corresponding homework exercises."
   ]
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     "text": [
      "0.0\n",
      "-7.6438855245442e-14\n"
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   ],
   "source": [
    "# just confirm that G.T@G and G@G.T are singluar for the 3x3 tomographyt example:\n",
    "G = np.array([\n",
    "    [1,1,1,0,0,0,0,0,0],\n",
    "    [0,0,0,1,1,1,0,0,0],\n",
    "    [0,0,0,0,0,0,1,1,1],\n",
    "    [1,0,0,1,0,0,1,0,0],\n",
    "    [0,1,0,0,1,0,0,1,0],\n",
    "    [0,0,1,0,0,1,0,0,1]])\n",
    "\n",
    "print(sl.det(G.T@G))\n",
    "print(sl.det(G@G.T))"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Summary\n",
    "\n",
    "\n",
    "- We've reviewed more linear algebra theory, especially as related to the rank of a matrix and how it relates to the solvability of linear systems\n",
    "\n",
    "\n",
    "- We've seen how we can use row operations to calculate the rank and to find the null-space of a matrix\n",
    "\n",
    "\n",
    "- Using geometric reasoning we've understood simple examples of problems with no solution, exactly one solution or infinitely many solutions in square and non-square cases\n",
    "\n",
    "\n",
    "- We've seen some examples of under- and over-determined problems and seen the solutions that can be obtained using the least squares and minimum norm analytical approach"
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