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   "source": [
    "# Assignment 4: Linear Algebra Review"
   ]
  },
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   "source": [
    "**1**. (20 points)\n",
    "\n",
    "Consider the linear transformation $f(x)$ on $\\mathbb{R}^3$ that takes the standard basis $\\left\\{e_1,e_2,e_3\\right\\}$ to $\\left\\{v_1,v_2,v_3\\right\\}$ where\n",
    "\n",
    "$$v_1=\\left(\\begin{matrix}10\\\\-10\\\\16\\end{matrix}\\right), v_2=\\left(\\begin{matrix}2\\\\-5\\\\20\\end{matrix}\\right) \\textrm {and } v_3=\\left(\\begin{matrix}1\\\\-4\\\\13\\end{matrix}\\right)$$\n",
    "\n",
    "1. Write a matrix $A$ that represents the same linear transformation. (4 points)\n",
    "\n",
    "2. Compute the rank of $A$ using two different methods (do not use `matrix_rank`!). (4 points)\n",
    "\n",
    "3. Find the eigenvalues and eigenvectors of $A$. (4 points)\n",
    "\n",
    "4. What is the matrix representation of $f$ with respect to the eigenbasis? (8 points)"
   ]
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    "\n",
    "\n",
    "\n"
   ]
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   "source": [
    "**2**. (20 points)\n",
    "\n",
    "You are given the following x-y coordinates (first column is x, second is y)\n",
    "\n",
    "```\n",
    "array([[  0.        ,   4.12306991],\n",
    "       [  3.        , -15.47355729],\n",
    "       [  4.        , -11.68725507],\n",
    "       [  3.        , -20.33756693],\n",
    "       [  5.        ,  -6.06401989],\n",
    "       [  6.        ,  32.79353057],\n",
    "       [  8.        ,  82.48658405],\n",
    "       [  9.        ,  84.02971858],\n",
    "       [  4.        ,  -1.30587276],\n",
    "       [  8.        ,  68.59409878]])\n",
    "```\n",
    "\n",
    "- Find the coefficients $(a, b, c)$ of the least-squares fit of a quadratic function $y = a + bx + cx^2$ to the data. \n",
    "- Plot the data and fitted curve using `matplotlib`.\n",
    "\n",
    "Note: Use `numpy.linalg.leastsq` function to solve this."
   ]
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   "source": [
    "\n",
    "\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**3**. (20 points)\n",
    "\n",
    "Consider the following system of equations:\n",
    "\n",
    "$$\\begin{align*}\n",
    "2x_1& - x_2&    +x_x    &=& 6\\\\\n",
    "-x_1& +2x_2& -  x_3 &=& 2\\\\\n",
    " x_1   &  -x_2& + x_3 &=& 1\n",
    "\\end{align*}$$\n",
    "\n",
    "1. Consider the system in matrix form $Ax=b$ and define $A$, $b$ in numpy.\n",
    "2. Show that $A$ is positive-definite\n",
    "3. Use the appropriate matrix decomposition function in numpy and back-substitution to solve the system. Remember to use the structure of the problem to determine the appropriate decomposition.\n",
    "\n",
    "\n"
   ]
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   "execution_count": null,
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   "outputs": [],
   "source": [
    "\n",
    "\n",
    "\n"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**4**. (40 points)\n",
    "\n",
    "You are given the following set of data to fit a quadratic polynomial to\n",
    "\n",
    "```python\n",
    "x = np.arange(10)\n",
    "y = np.array([  1.58873597,   7.55101533,  10.71372171,   7.90123225,\n",
    "                -2.05877605, -12.40257359, -28.64568712, -46.39822281,\n",
    "                -68.15488905, -97.16032044])\n",
    "```\n",
    "\n",
    "- Find the least squares solution by using the normal equations $A^T A \\hat{x} = A^T y$. (5 points)\n",
    "\n",
    "- Write your own **gradient descent** optimization function to find the least squares solution for the coefficients $\\beta$ of a quadratic polynomial. Do **not** use a gradient descent algorithm from a package such as `scipy-optimize` or `scikit-learn`.  You can use a simple for loop - start with the parameters `beta = np.zeros(3)` with a learning rate $\\alpha = 0.0001$ and run for 100000 iterations. (15 points)\n",
    "\n",
    "- Plot the data together with the fitted polynomial. (10 points)\n",
    "\n"
   ]
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  {
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   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\n",
    "\n"
   ]
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