{
"cells": [
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"import matplotlib as mpl\n",
"import numpy as np\n",
"from mpl_toolkits.mplot3d import Axes3D\n",
"import scipy as sp\n",
"import scipy.linalg\n",
"import scipy.spatial\n",
"import sympy as sy\n",
"sy.init_printing() "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# The Dot Product"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Consider two vectors\n",
"\n",
"$$\n",
"\\mathbf{u}=\\left[\\begin{array}{l}\n",
"u_{1} \\\\\n",
"u_{2} \\\\\n",
"\\vdots \\\\\n",
"u_{n}\n",
"\\end{array}\\right] \\quad \\text { and } \\bf{v}=\\left[\\begin{array}{l}\n",
"v_{1} \\\\\n",
"v_{2} \\\\\n",
"\\vdots \\\\\n",
"v_{n}\n",
"\\end{array}\\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The dot product of $\\mathbf{u}$ and $\\mathbf{v}$,i.e. $\\mathbf{u}\\cdot\\mathbf{v}$ is defined as\n",
"\n",
"$$\n",
"\\left[\\begin{array}{llll}\n",
"u_{1} & u_{2} & \\cdots & u_{n}\n",
"\\end{array}\\right]\\left[\\begin{array}{c}\n",
"v_{1} \\\\\n",
"v_{2} \\\\\n",
"\\vdots \\\\\n",
"v_{n}\n",
"\\end{array}\\right]=u_{1} v_{1}+u_{2} v_{2}+\\cdots+u_{n} v_{n}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can generate two random vectors, then let's compare operations in NumPy."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"u = np.round(100*np.random.randn(10))\n",
"v = np.round(100*np.random.randn(10))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ -846., -1700., -6486., -1886., -1100., 6776., -6210., 1586.,\n",
" -245., 2394.])"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u*v # this is element-wise multiplication"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle -7717.0$"
],
"text/plain": [
"-7717.0"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u@v # matrix multiplication"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle -7717.0$"
],
"text/plain": [
"-7717.0"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.inner(u,v) # inner product here is the same as matrix multiplication"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"SymPy operation is the same like in linear algebra."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"u = sy.randMatrix(1, 10)\n",
"v = sy.randMatrix(1, 10)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}15991\\end{matrix}\\right]$"
],
"text/plain": [
"[15991]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u*v.T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# The Norm of a Vector"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The norm is the length of a vector, definted by\n",
"\n",
"$$\n",
"\\|\\mathbf{v}\\|=\\sqrt{\\mathbf{v} \\cdot \\mathbf{v}}=\\sqrt{v_{1}^{2}+v_{2}^{2}+\\cdots+v_{n}^{2}}, \\quad \\text { and } \\quad\\|\\mathbf{v}\\|^{2}=\\mathbf{v} \\cdot \\mathbf{v}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A NumPy built-in ```np.linalg.norm()``` is for computing norms.The default setting is to compute the length of vectors from origins."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 6.324555320336759$"
],
"text/plain": [
"6.324555320336759"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"a = [2, 6]\n",
"np.linalg.norm(a)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Verify the results."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 6.324555320336759$"
],
"text/plain": [
"6.324555320336759"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.sqrt(2**2 + 6**2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can also compute a group of vectors' length, for instance $(2, 6)^T$, $(8, 2)^T$, $(9, 1)^T$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([6.32455532, 8.24621125, 9.05538514])"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = np.array([[2, 8, 9], \n",
" [6, 2, 1]])\n",
"np.linalg.norm(A, axis = 0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Distance in $\\mathbb{R}^n$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For $\\mathbf{u}$ and $\\mathbf{v}$ in $\\mathbb{R}^{n}$, the distance between $\\mathbf{u}$ and $\\mathbf{v},$ written as dist $(\\mathbf{u}, \\mathbf{v}),$ is the length of the vector $\\mathbf{u}-\\mathbf{v} .$ That is,\n",
"\n",
"$$\n",
"\\operatorname{dist}(\\mathbf{u}, \\mathbf{v})=\\|\\mathbf{u}-\\mathbf{v}\\|\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Suppose we have two vectors $\\mathbf{u} = (2, 9)$ and $\\mathbf{v} = (-3, 4)$, compute the distance and visualize the results."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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\n",
"text/latex": [
"$\\displaystyle 7.0710678118654755$"
],
"text/plain": [
"7.0710678118654755"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u = np.array([2, 9])\n",
"v = np.array([-3, 4])\n",
"np.linalg.norm(u - v)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
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