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    "---\n",
    "title: Biostat 216 Homework 4\n",
    "subtitle: 'Due Nov 10 @ 11:59pm'\n",
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    "Submit a PDF (scanned/photographed from handwritten solutions, or converted from RMarkdown or Jupyter Notebook or Quarto) to Gradescope in BruinLearn. \n",
    "\n",
    "## Q1\n",
    "\n",
    "For any $\\mathbf{X} \\in \\mathbb{R}^{n \\times p}$ and $\\mathbf{y} \\in \\mathbb{R}^n$, show that the normal equation \n",
    "$$\n",
    "\\mathbf{X}' \\mathbf{X} \\boldsymbol{\\beta} = \\mathbf{X}' \\mathbf{y}\n",
    "$$\n",
    "always has at least one solution.\n",
    "\n",
    "## Q2\n",
    "\n",
    "Let $\\mathbf{A} \\in \\mathbb{R}^{m \\times n}$.\n",
    "\n",
    "1. Show that for any generalized inverse $\\mathbf{A}^-$, we have $\\text{rank}(\\mathbf{A}^-) \\ge \\text{rank}(\\mathbf{A})$.\n",
    "\n",
    "2. Show that the Moore-Penrose inverse $\\mathbf{A}^+$ has the same rank as $\\mathbf{A}$.\n",
    "\n",
    "## Q3 Householder reflections\n",
    "\n",
    "Let $\\mathbf{v} \\in \\mathbb{R}^n$. Define the **Householder reflection matrix**\n",
    "$$\n",
    "\\mathbf{H} = \\mathbf{I} - 2 \\frac{\\mathbf{v} \\mathbf{v}'}{\\|\\mathbf{v}\\|^2} = \\mathbf{I} - 2 \\mathbf{u} \\mathbf{u}',\n",
    "$$\n",
    "where $\\mathbf{u}$ is the unit vector $\\mathbf{v} / \\|\\mathbf{v}\\|$.\n",
    "\n",
    "1. Show that $\\mathbf{H}$ is symmetric and orthogonal.\n",
    "\n",
    "2. Let $\\mathbf{a}, \\mathbf{b} \\in \\mathbb{R}^n$ such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\|$. Find a Householder matrix such that $\\mathbf{H} \\mathbf{a} = \\mathbf{b}$.\n",
    "\n",
    "3. Let $\\mathbf{a} \\in \\mathbb{R}^n$ be a non-zero vector. Find a Householder matrix such that\n",
    "$$\n",
    "\\mathbf{H} \\mathbf{a} = \\begin{pmatrix} \\|\\mathbf{a}\\| \\\\ \\mathbf{0}_{n-1} \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "4. Let $\\mathbf{a} \\in \\mathbb{R}^n$. Find a Householder matrix such that\n",
    "$$\n",
    "\\mathbf{H} \\mathbf{a} = \\begin{pmatrix} a_1 \\\\ \\|\\mathbf{a}_{2:n}\\| \\\\ \\mathbf{0}_{n-2} \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "5. Let $\\mathbf{A} \\in \\mathbb{R}^{n \\times p}$. Describe how to find a sequence of Householder matrices $\\mathbf{H}_1, \\ldots, \\mathbf{H}_{p}$ such that\n",
    "$$\n",
    "\\mathbf{H}_{p} \\mathbf{H}_{p-1} \\cdots \\mathbf{H}_1 \\mathbf{A} = \\mathbf{R},\n",
    "$$\n",
    "where $\\mathbf{R} \\in \\mathbb{R}^{n \\times p}$ is an upper triangular matrix.  \n",
    "\n",
    "    Describe how this generates a full QR decomposition of matrix $\\mathbf{A} = \\mathbf{Q} \\mathbf{R}$, where $\\mathbf{Q} \\in \\mathbb{R}^{n \\times n}$ is an orthogonal matrix and $\\mathbf{R}$ is upper triangular.\n",
    "\n",
    "## Q4 Missile tracking\n",
    "\n",
    "A missile is fired from enemy territory, and its position in flight is observed by radar tracking devices at the following positions.\n",
    "\n",
    "| $x$=Position down range (1000 miles) | 0 |  0.25 |  0.5  |  0.75 |   1   |\n",
    "|:------------------------------------:|:-:|:-----:|:-----:|:-----:|:-----:|\n",
    "|        $y$=Height (1000 miles)       | 0 | 0.008 | 0.015 | 0.019 | 0.020 |\n",
    "\n",
    "Suppose that intelligence sources indicate that enemy missiles are programmed to follow a parabolic flight path: $y = f(x) = \\alpha_0 + \\alpha_1 x + \\alpha_2 x^2$. Where is the missile expected to land? Hint: You can find the solution using computer program. For example, in Julia, least squares solution is obtained by command `A \\ b`.\n",
    "\n",
    "![](missile.png)\n",
    "\n",
    "## BV exercises\n",
    "\n",
    "12.2, 12.3"
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