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   "source": [
    "## CS 132 Final Exam\n",
    "\n",
    "### Due Sunday August 15th at noon\n",
    "\n",
    "\n",
    "Please complete 10 of the following 12 problems; for those you are omitting, please write \"SKIP\" in\n",
    "the solution cell. If you do not skip 2 problems, or if I can not figure out your intentions, \n",
    "I will delete problems 11 and 12. \n",
    "\n",
    "You may NOT skip parts of problems: If you choose to do a problem, you must do all parts for full credit; if\n",
    "you skip a problem, no part of that problem will count. \n",
    "\n",
    "You may use any materials from the course, and may, of course, use Python to calculate your\n",
    "results. You may consult Python documentation,  if you need to find the correct library\n",
    "or the syntax for, say, the `arccos` function. But you may NOT Google around to find the precise answer to a question, and you may NOT consult with another person in developing your answer.\n",
    "\n",
    "Richard and I will be monitoring Piazza regularly during the exam period: post questions there\n",
    "if you need advice. \n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Here are some imports which will be used in the code in the rest of the lab  \n",
    "\n",
    "# Imports used for the code in CS 237\n",
    "\n",
    "import numpy as np                # arrays and functions which operate on arrays\n",
    "\n",
    "import matplotlib.pyplot as plt   # normal plotting\n",
    "\n",
    "\n",
    "# NOTE: You may not use any other libraries than those listed here without explicit permission."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem One (T/F)\n",
    "\n",
    "This problem consists of 10 statements. For each, state whether it is true or false and give a brief, one-sentence justification. If possible, when the statement is false, give a counter-example as justification.  \n",
    "\n",
    "(1) It is possible for a set of $n$ vectors to span all of $\\mathbb{R}^m$ when $n < m$.\n",
    "\n",
    "(2) If an $n\\times m$ matrix $A$ has less than $n$ pivots when reduced to Reduced Echelon Form, then it can not span\n",
    "all of $\\mathbb{R}^n$.\n",
    "\n",
    "(3) The homogeneous equation $A{\\bf x} = {\\bf 0}$ has the trivial solution\n",
    "if and only if the equation has at least one free\n",
    "variable.\n",
    "\n",
    "(4) Suppose $A$ is a $3 \\times 3$ matrix and $\\bf b$ is a vector in $\\mathbb{R}^3$ such that\n",
    "the equation $A{\\bf x} = {\\bf b}$ does not have a solution. Then there\n",
    "exists a vector ${\\bf y}$ in $\\mathbb{R}^3$ such that the equation $A{\\bf x} = {\\bf y}$ has a\n",
    "unique solution.\n",
    "\n",
    "(5) If $S$ is a linearly dependent set of vectors, then each vector is a linear\n",
    "combination of the *other* vectors in $S.$\n",
    "\n",
    "(6) If vectors $\\{v_1, v_2,v_3, v_4\\}$ are linearly independent vectors in $\\mathbb{R}^4$, then\n",
    "$\\{v_1, v_2,v_3, (v_3+v_4)\\}$ is also linearly independent.\n",
    "\n",
    "(7) Suppose $T({\\bf x}) = A{\\bf x} - {\\bf b},$ with $A$ an $m \\times n$ matrix and ${\\bf b} \\in\\mathbb{R}^m$. \n",
    "$T$ is a linear transformation if and only if  ${\\bf b}= \\bf 0$. \n",
    "\n",
    "(8) Let $T : \\mathbb{R}^n \\rightarrow \\mathbb{R}^m$ be a linear transformation, and let $A$ be the standard matrix for $T$ . Then: $T$ is one-to-one if and only if the columns of $A$ are linearly independent.\n",
    "\n",
    "(9) If a system of linear equations has two different solutions,\n",
    "it must have infinitely many solutions.\n",
    "\n",
    "(10) If $A$ is invertible, then the columns of $A^{-1}$ are linearly\n",
    "independent.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Two (Short Answer)\n",
    "\n",
    "Give brief answers to the following questions. If the answer is a number, give a brief justification. \n",
    "\n",
    "\n",
    "(1) Suppose the last column of $AB$ consists of all 0's but $B$ itself has no column of zeros. Are the columns of\n",
    "$A$ linearly dependent or linearly independent?  Explain why briefly. \n",
    "\n",
    "(2) Let $T : \\mathbb{R}^2\\rightarrow \\mathbb{R}^2$ be a linear transformation that first rotates a point (about the origin) through $-\\frac{3}{4}\\pi$ radians\n",
    "(i.e., clockwise) and then reflects it through the horizontal $x$-axis. Find the standard matrix of $T$.\n",
    "\n",
    "(3) Let $A =\\begin{bmatrix} 2 & 5 \\\\ -3 & 1     \\end{bmatrix}$ and $B =\\begin{bmatrix} 4 & -5 \\\\ 3 & k     \\end{bmatrix}$. What value(s) of $k$, if any, will make $AB = BA$?\n",
    "\n",
    "(4) What is the rank of a $5\\times 6$ matrix whose null space is three-dimensional?\n",
    "\n",
    "(5) Consider a $3\\times 3$ matrix $A$ with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable?\n",
    "Why or why not?\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Three\n",
    "\n",
    "Consider the following matrix:\n",
    "\n",
    "![Screen%20Shot%202021-08-11%20at%2012.08.28%20PM.png](attachment:Screen%20Shot%202021-08-11%20at%2012.08.28%20PM.png)\n",
    "\n",
    "\n",
    "(A)  Show that the columns of A are orthogonal by an appropriate multiplication of two matrices. Don’t\n",
    "multiply any individual vectors and don't use Gaussian Elimination! Use Python, and show a trace of your computation.  \n",
    "\n",
    "\n",
    "(B) Does $A^T$ have orthogonal columns? Show how you test this in Python, and answer the question.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-08-11%20at%2011.48.29%20AM.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Four\n",
    "\n",
    "Consider the following matrix $A$:\n",
    "\n",
    "![Screen%20Shot%202021-08-11%20at%2011.48.29%20AM.png](attachment:Screen%20Shot%202021-08-11%20at%2011.48.29%20AM.png)\n",
    "\n",
    "\n",
    "For each of the four fundamental subspaces of $A$, give the dimensionality of the subspace and calculate\n",
    "a basis for that subspace (you may use your code from homeworks to help with this). \n",
    "\n",
    "(A)  $Col(A)$\n",
    "\n",
    "(B)  $Nul(A)$\n",
    "\n",
    "(C)  $Col(A^T)$  (also called the \"row space of A\")\n",
    "\n",
    "(D)  $Nul(A^T)$   (also called the \"left null space of A\")  \n",
    "\n",
    "Now,  again using code from homeworks or any other technique you prefer, show that \n",
    "\n",
    "(E)  $Nul(A^T)$ is the orthogonal complement of $Col(A)$\n",
    "\n",
    "(F) $Nul(A)$ is the orthogonal complement of $Col(A^T)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Five\n",
    "\n",
    "Let $A$ be a *lower triangular* $n\\times n$ matrix with nonzero entries on the diagonal. Show that $A$ is invertible and\n",
    "that $A^{-1}$ is lower triangular. \n",
    "\n",
    "Hint: Think about the method given in lecture for constructing the inverse of a matrix. \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Six\n",
    "\n",
    "Consider the infinite set of vectors in $\\mathbb{R}^4$ of the form $\\begin{bmatrix} a\\\\-a\\\\b\\\\0 \\end{bmatrix}$\n",
    "for any real numbers $a$ and $b$. \n",
    "\n",
    "\n",
    "(A) What is the geometrical shape described by this set, i.e., how many dimensions?\n",
    "\n",
    "(B) Is this set a subspace of $\\mathbb{R}^4$? Explain why or why not. \n",
    "\n",
    "(C) Does this set span all of $\\mathbb{R}^n$ for some $n$?  Explain why or why not. \n",
    "\n",
    "\n",
    "(D) If the set is a subspace give a basis for the set. \n",
    "                             "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Seven\n",
    "\n",
    "Suppose $A = UDV^T$ where $U$ and $V$ are $n\\times n$ matrices with the property that $U^TU = I$ and $V^TV = I,$ and\n",
    "where $D$ is a diagonal matrix where all the values are greater than 0. \n",
    "\n",
    "(A) Explain why $D$ is invertible, \n",
    "\n",
    "(B) Show that $A$ is invertible, and\n",
    "\n",
    "(C) Find a formula for $A^{-1}.$ \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n"
   ]
  },
  {
   "attachments": {
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IJgJEXplAUTX9IUDk1V9MyCOZCBB5ZQJF1fSHwP8BofgfIHA4cOkAAAAASUVORK5CYII="
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Eight\n",
    "\n",
    "Diagonalize this matrix, whose eigenvalues are 2 and 8. Note: You only need to find $P$ and $D$. Adjust\n",
    "the \n",
    "eigenvectors (by scaling) so that they  have a 1 in the lowest nonzero position.\n",
    "\n",
    "![Screen%20Shot%202021-08-11%20at%2012.03.57%20PM.png](attachment:Screen%20Shot%202021-08-11%20at%2012.03.57%20PM.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:** "
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-07-02%20at%2010.02.09%20AM.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Nine\n",
    "\n",
    "(A)  Suppose $\\bf y$ is orthogonal to both $\\bf u$ and $\\bf v.$ Show that $\\bf y$ is orthogonal to every $\\bf w$ in $Span({\\bf u},{\\bf v})$. Remember\n",
    "that every $\\bf w$ in $Span({\\bf u},{\\bf v})$ can be written in the form $c_1{\\bf u}+c_2{\\bf v}$.\n",
    "\n",
    "Consider the following vectors:\n",
    "\n",
    "![Screen%20Shot%202021-07-02%20at%2010.02.09%20AM.png](attachment:Screen%20Shot%202021-07-02%20at%2010.02.09%20AM.png)\n",
    "\n",
    "\n",
    "\n",
    "(B) Show that $\\{ {\\bf u}_1, {\\bf u}_2,{\\bf u}_3 \\}$ is an orthogonal basis for $\\mathbb{R}^3$.\n",
    "\n",
    "\n",
    "(C) Express $\\bf x$ as a linear\n",
    "combination of the ${\\bf u}$’s."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-07-02%20at%209.15.36%20AM.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Ten\n",
    "\n",
    "Given $A$ whose factorization $PKP^{-1}$ is as shown below, find the eigenvalues of $A$ and a basis for each eigenspace. \n",
    "\n",
    "![Screen%20Shot%202021-07-02%20at%209.15.36%20AM.png](attachment:Screen%20Shot%202021-07-02%20at%209.15.36%20AM.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Eleven\n",
    "\n",
    "Suppose a non-homogeneous system of 5 linear equations in 7 unknowns has a solution, with two\n",
    "free variables. Is it possible to change some constants on the equations’ right sides to make the new system\n",
    "inconsistent? Justify your answer, using the properties of the Null space, Column space, and the Rank Theorem. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problem Twelve\n",
    "\n",
    "Assume an $n\\times n$ matrix $A$is diagonalizable, so $A = PDP^{-1}$ for some $P,$ with $D$ a diagonal matrix\n",
    "with $\\lambda_1, \\lambda_2, \\ldots, \\lambda_n$ along the diagonal.  Assume\n",
    "$A$ is invertible, and let $B$ be the inverse of $A.$ It happens that $A$ and $B$ have the same eigenvectors (in fact, a matrix\n",
    "and its inverse always have the same eigenvectors). Find a formula for the inverse of A, making use of P and D\n",
    "as necessary."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  }
 ],
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