auto_quizzes/roch-mmids-ls-autoquiz.ipynb

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   "source": [
    "***\n",
    "\n",
    "*Course:* [Math 535](https://people.math.wisc.edu/~roch/mmids/) - Mathematical Methods in Data Science (MMiDS)  \n",
    "*Chapter:* 2-Least squares   \n",
    "*Author:* [Sebastien Roch](https://people.math.wisc.edu/~roch/), Department of Mathematics, University of Wisconsin-Madison  \n",
    "*Updated:* Jan 4, 2024   \n",
    "*Copyright:* &copy; 2024 Sebastien Roch\n",
    "\n",
    "***"
   ]
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   "source": [
    "## Auto-quizzes"
   ]
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   "source": [
    "This notebook generates automated quizzes as well as the answers. Set the `seed` to any integer to produce unique quizzes."
   ]
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   "source": [
    "# PYTHON 3\n",
    "import numpy as np\n",
    "from numpy import linalg as LA\n",
    "from numpy.random import default_rng"
   ]
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   "source": [
    "# Set the `seed` to any integer\n",
    "seed=535"
   ]
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   "source": [
    "rng = default_rng(seed)"
   ]
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   "source": [
    "**AQ2.1**  \n",
    "\n",
    "***\n",
    "\n",
    "*Use the following code to generate the quiz questions. You should be able to answer them by hand -- that is, without the help of numerical computation.*\n",
    "\n",
    "***"
   ]
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   "source": [
    "Consider the following matrix $A$:"
   ]
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   "source": [
    "A = np.sign(2 * rng.random(size=(3,2)) - 1)\n",
    "i = rng.integers(low=0,high=3,size=1)\n",
    "A[i,0] = 0\n",
    "print(A)"
   ]
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   "source": [
    "and the following column vector $\\mathbf{z}$:"
   ]
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   "source": [
    "z = rng.integers(low=-2,high=2,size=3)\n",
    "print(z)"
   ]
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   "source": [
    "(a) What is the rank of $A$?\n",
    "\n",
    "(b) Give a QR decomposition of $A$ where the first column of $Q$ is a multiple of the first column of $A$.\n",
    "\n",
    "(c) Compute the orthogonal projection of $\\mathbf{z}$ onto the column space of $A$.\n",
    "\n",
    "(d) Complete $Q$ (i.e. add a column to it) into an orthogonal matrix $W$."
   ]
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   "source": [
    "***\n",
    "\n",
    "*Use the following code to generate the answers.*\n",
    "\n",
    "***"
   ]
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   "source": [
    "# (a)\n",
    "LA.matrix_rank(A)"
   ]
  },
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   "source": [
    "# (b) [Note: Multiypling either column of Q by -1 is also a valid answer.]\n",
    "Q, R = LA.qr(A)\n",
    "print(Q)"
   ]
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   "source": [
    "print(R)"
   ]
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   "source": [
    "# (c)\n",
    "proj_z = Q @ Q.T @ z\n",
    "print(proj_z)"
   ]
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   "source": [
    "# (d) [Note: Multiplying any of the columns by -1 also gives a valid answer.]\n",
    "p = np.resize((z - proj_z) / LA.norm(z - proj_z), (3,1))\n",
    "W = np.hstack((Q,p))\n",
    "print(W)"
   ]
  },
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   },
   "source": [
    "$\\lhd$"
   ]
  }
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