{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CS 132 Homework 02 Solution -- \n",
    "\n",
    "### Due Wednesday July 21st at Midnight (1 minute after 11:59pm) in Gradescope (with grace period of 6 hours)\n",
    "### Homeworks may be submitted up to 24 hours late with a 10% penalty (same grace period)\n",
    "\n",
    "\n",
    "Please read through the entire notebook, reading the expository material and then do the problems, following the instuctions to try various features; among the exposition of various features of Python there\n",
    "are three problems which will be graded.  All problems on homeworks are worth 10 points towards your final homework score (60% of your final grade). \n",
    "\n",
    "You will need to complete this notebook and then convert to a PDF file in order to submit to Gradescope. Instructions are given here:\n",
    "\n",
    "https://www.cs.bu.edu/fac/snyder/cs132/HWSubmissionInstructions.html\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "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 1 (Python Programming)\n",
    "\n",
    "For this problem, you must implement a set of functions in Python to analyze and solve linear systems. \n",
    "\n",
    "Solving a linear system using Gaussian Elimination requires implementing three functions:\n",
    "\n",
    "1. forward elmination (Stage 1)\n",
    "2. testing for consistency\n",
    "3. backsubstitution (Stage 2)\n",
    "\n",
    "We provide the code below for forward elimination. Your job is to provide the rest of the code and use the resulting\n",
    "set of functions to analyze some linear systems. (In case it helps, the code you have to write is (quite a bit)\n",
    "shorter than the code we have already supplied!)\n",
    "\n",
    "In the  code cell below we have provided the function `forwardElimination(B)`\n",
    "which takes a matrix (a numpy array) $B$ and returns a new matrix that is the row echelon form of $B.$ \n",
    "\n",
    "### What you need to do\n",
    "\n",
    "You need two write two functions:\n",
    "\n",
    "      inconsistentSystem(B)\n",
    "      \n",
    "which takes a matrix $B$ in echelon form, and returns `True` if it represents an inconsistent system, and `False`\n",
    "otherwise; and\n",
    "\n",
    "      backsubstitution(B)\n",
    "\n",
    "which returns the reduced row echelon form matrix of $B$.\n",
    "\n",
    "Using these, you should be able to analyze a matrix representing a linear system as follows:\n",
    "         \n",
    "    AEchelon = forwardElimination(A)\n",
    "    if (not inconsistentSystem(AEchelon)):\n",
    "        AReducedEchelon = backsubstitution(AEchelon)\n",
    "        print(AReducedEchelon)\n",
    "    else:\n",
    "        print(\"System is inconsistent!\")\n",
    "         \n",
    "and then by inspecting `AReducedEchelon` you should be able to write down the solution set for the linear\n",
    "system.\n",
    "\n",
    "**Hints.** \n",
    "First, I encourage you to read carefully the code that is already provided before starting to write your own\n",
    "code. Most of what you need to do can be patterned after code that is already provided, as long as you understand\n",
    "what the provided code is doing!\n",
    "\n",
    "The second thing to do is to look at the `numpy` tutorial in the tutorials directory on the class web page.\n",
    "In it, you can see how to do useful things like multiply a row of a matrix by a float, and other operations you might need. \n",
    "\n",
    "One of the main things to notice is that there is a function provided for you that does row reduction. If you look\n",
    "at this function you will see that it specifically handles floating point inaccuracies. So if you need to do row\n",
    "reduction in your code, you should use that function rather than coding something else yourself.\n",
    "Here is my #1 piece of advice for success on this and future programming assignments: work incrementally,\n",
    "writing and testing small bits of code at a time. Don’t implement the whole thing and then try to debug it; that\n",
    "will take much more time than if you break the process into smaller pieces and make sure they work at each\n",
    "stage.\n",
    "\n",
    "\n",
    "Additionally, you are to run your functions on seven problems.\n",
    "\n",
    "(A)  First, use your code to solve the following linear system:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "        x_1+2x_2+3x_3&= -16  \\\\       \n",
    "        5x_1+4x_2+6x_3 &= -41 \\\\\n",
    "        10x_1+9x_2+8x_3 &= -50  \\\\ \n",
    "  \\end{aligned}$$\n",
    "  \n",
    "Report the solution set.\n",
    "\n",
    "(B) Second, use your code to analyze the six linear systems found in the files h2m1.txt, h2m2.txt,\n",
    "h2m3.txt, h2m4.txt, h2m5.txt and h2m6.txt. These are loaded for you in the last code cell. \n",
    "\n",
    "\n",
    "For each system, state whether the system is\n",
    "consistent, and if so, write down the solution set.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import warnings\n",
    "\n",
    "def swapRows(A, i, j):\n",
    "    \"\"\"\n",
    "    interchange two rows of A\n",
    "    operates on A in place\n",
    "    \"\"\"\n",
    "    tmp = A[i].copy()\n",
    "    A[i] = A[j]\n",
    "    A[j] = tmp\n",
    "\n",
    "def relError(a, b):\n",
    "    \"\"\"\n",
    "    compute the relative error of a and b\n",
    "    \"\"\"\n",
    "    with warnings.catch_warnings():\n",
    "        warnings.simplefilter(\"error\")\n",
    "        try:\n",
    "            return np.abs(a-b)/np.max(np.abs(np.array([a, b])))\n",
    "        except:\n",
    "            return 0.0\n",
    "\n",
    "def rowReduce(A, i, j, pivot):\n",
    "    \"\"\"\n",
    "    reduce row j using row i with pivot pivot, in matrix A\n",
    "    operates on A in place\n",
    "    \"\"\"\n",
    "    factor = A[j][pivot] / A[i][pivot]\n",
    "    for k in range(len(A[j])):\n",
    "        # we allow an accumulation of error 100 times larger than a single computation\n",
    "        # this is crude but works for computations without a large dynamic range\n",
    "        if relError(A[j][k], factor * A[i][k]) < 100 * np.finfo('float').resolution:\n",
    "            A[j][k] = 0.0\n",
    "        else:\n",
    "            A[j][k] = A[j][k] - factor * A[i][k]\n",
    "\n",
    "# stage 1 (forward elimination)\n",
    "def forwardElimination(B):\n",
    "    \"\"\"\n",
    "    Return the row echelon form of B\n",
    "    \"\"\"\n",
    "    A = B.copy().astype(float)\n",
    "    m, n = np.shape(A)\n",
    "    for i in range(m-1):\n",
    "        # Let A[leftmostNonZeroCRow][leftmostNonZeroCol] be the leftmost nonzero value \n",
    "        # in row i or any row below it; this will be the pivot value for row i, assuming\n",
    "        # such exists\n",
    "        leftmostNonZeroRow = m    # sentinel values\n",
    "        leftmostNonZeroCol = n\n",
    "        ## for each row at or below row i\n",
    "        for h in range(i,m):\n",
    "            ## search, starting from the left, for the first nonzero in row h\n",
    "            for k in range(i,n):\n",
    "                if (A[h][k] != 0.0) and (k < leftmostNonZeroCol):\n",
    "                    leftmostNonZeroRow = h\n",
    "                    leftmostNonZeroCol = k\n",
    "                    break\n",
    "        # if there is no such position, stop\n",
    "        if leftmostNonZeroRow == m:\n",
    "            break\n",
    "        # If the leftmostNonZeroCol in row i is zero, swap this row \n",
    "        # with a row below it\n",
    "        # to make that position nonzero. This creates a pivot in that position.\n",
    "        if (leftmostNonZeroRow > i):\n",
    "            swapRows(A, leftmostNonZeroRow, i)\n",
    "        # Use row reduction operations to create zeros in all positions \n",
    "        # below the pivot.\n",
    "        for h in range(i+1,m):\n",
    "            rowReduce(A, i, h, leftmostNonZeroCol)\n",
    "    return A\n",
    "\n",
    "#################### \n",
    "\n",
    "# If any operation creates a row that is all zeros except the last element,\n",
    "# the system is inconsistent; stop.\n",
    "def inconsistentSystem(A):\n",
    "    \"\"\"\n",
    "    B is assumed to be in echelon form; return True if it represents\n",
    "    an inconsistent system, and False otherwise\n",
    "    \"\"\"\n",
    "    \n",
    "    # Your code here\n",
    "        \n",
    "    return False\n",
    "\n",
    "def backsubstitution(B):\n",
    "    \"\"\"\n",
    "    return the reduced row echelon form matrix of B\n",
    "    \"\"\"\n",
    "    \n",
    "    # Your code here\n",
    "    \n",
    "    return A\n",
    "\n",
    "\n",
    "#####################\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solution (A)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solution set:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[5.76402095e-01 5.18898527e-01 7.78899193e-01 1.57304293e-01\n",
      "  8.69013675e-01 9.97183090e-01 2.09838914e+01]\n",
      " [3.77557697e-01 8.52586464e-01 5.56484979e-01 7.69544094e-01\n",
      "  1.79446494e-01 9.23742939e-01 2.18819108e+01]\n",
      " [6.85388721e-01 8.60473973e-01 6.94989389e-01 6.01698698e-01\n",
      "  2.42080170e-01 1.73404027e-01 1.55698084e+01]\n",
      " [2.94221629e-01 7.63110260e-01 9.48824073e-01 5.92042316e-02\n",
      "  9.01763148e-01 6.08513454e-01 1.86695329e+01]\n",
      " [7.39209796e-01 3.21291282e-01 1.31851468e-01 4.28161407e-01\n",
      "  9.34928120e-01 1.92181367e-01 1.46766210e+01]\n",
      " [8.76228244e-01 8.68739630e-01 3.27679890e-01 2.02894777e-02\n",
      "  3.17427217e-01 9.74364614e-01 1.64024788e+01]] \n",
      "\n",
      "[[5.76402095e-01 5.18898527e-01 7.78899193e-01 1.57304293e-01\n",
      "  8.69013675e-01 9.97183090e-01 2.09838914e+01]\n",
      " [3.77557697e-01 8.52586464e-01 5.56484979e-01 7.69544094e-01\n",
      "  1.79446494e-01 9.23742939e-01 2.18819108e+01]\n",
      " [7.39209796e-01 3.21291282e-01 1.31851468e-01 4.28161407e-01\n",
      "  9.34928120e-01 1.92181367e-01 1.56766210e+01]\n",
      " [2.94221629e-01 7.63110260e-01 9.48824073e-01 5.92042316e-02\n",
      "  9.01763148e-01 6.08513454e-01 1.86695329e+01]\n",
      " [7.39209796e-01 3.21291282e-01 1.31851468e-01 4.28161407e-01\n",
      "  9.34928120e-01 1.92181367e-01 1.46766210e+01]\n",
      " [8.76228244e-01 8.68739630e-01 3.27679890e-01 2.02894777e-02\n",
      "  3.17427217e-01 9.74364614e-01 1.64024788e+01]] \n",
      "\n",
      "[[0.97207478 0.90673997 0.73327347 0.52275659 0.85977086 3.55204977]\n",
      " [0.86745881 0.7722276  0.79776925 0.09137848 0.42059553 4.2778121 ]\n",
      " [0.03179614 0.76143713 0.30113213 0.44570075 0.90351831 0.67299846]\n",
      " [0.39951036 0.36724984 0.14290639 0.65431492 0.8181544  0.47335011]\n",
      " [0.40108651 0.59150112 0.0838722  0.28653963 0.75340726 1.06872286]\n",
      " [0.03206643 0.52664998 0.63187483 0.11303681 0.23847471 1.80160913]] \n",
      "\n",
      "[[ 7. -1. -6.  2.  4. -8.]\n",
      " [-5.  5.  4.  8. 16. -4.]\n",
      " [ 3.  2.  0.  1.  1.  2.]] \n",
      "\n",
      "[[  0.94571468   0.88312203   0.49213102 -34.61964818]\n",
      " [  0.31011985   0.62534437   0.27655419 -13.29511081]\n",
      " [  0.25685932   0.16001878   0.20955331 -17.30629434]\n",
      " [  0.89584391   0.78032557   0.6655545  -50.18041106]\n",
      " [  0.66052984   0.55214113   0.09730802  -5.6427922 ]\n",
      " [  0.31913839   0.60976324   0.65022581 -44.04057531]\n",
      " [  0.79098811   0.3397772    0.36264537 -33.7639984 ]\n",
      " [  0.6527187    0.07032046   0.26754685 -29.91502505]] \n",
      "\n",
      "[[ 0.3862021   0.86035917  0.09708003  0.36647638  0.49856039  0.48273141]\n",
      " [ 0.73894056  0.21447831  0.42631758  0.6148471   0.58543139  1.90369738]\n",
      " [ 0.01778327  0.52049088  0.43152589  0.65779392  0.01553748 -0.90863122]] \n",
      "\n"
     ]
    }
   ],
   "source": [
    "h2m1 = np.loadtxt('https://www.cs.bu.edu/fac/snyder/cs132/Homeworks/h2m1.txt')\n",
    "h2m2 = np.loadtxt('https://www.cs.bu.edu/fac/snyder/cs132/Homeworks/h2m2.txt')\n",
    "h2m3 = np.loadtxt('https://www.cs.bu.edu/fac/snyder/cs132/Homeworks/h2m3.txt')\n",
    "h2m4 = np.loadtxt('https://www.cs.bu.edu/fac/snyder/cs132/Homeworks/h2m4.txt')\n",
    "h2m5 = np.loadtxt('https://www.cs.bu.edu/fac/snyder/cs132/Homeworks/h2m5.txt')\n",
    "h2m6 = np.loadtxt('https://www.cs.bu.edu/fac/snyder/cs132/Homeworks/h2m6.txt')\n",
    "\n",
    "print(h2m1,'\\n')\n",
    "print(h2m2,'\\n')\n",
    "print(h2m3,'\\n')\n",
    "print(h2m4,'\\n')\n",
    "print(h2m5,'\\n')\n",
    "print(h2m6,'\\n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solution (B)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 2\n",
    "\n",
    "Chemical equations describe the quantities of substances consumed and produced by\n",
    "chemical reactions. For instance, when propane gas burns, the propane ($C_3H_8$) combines\n",
    "with oxygen ($O_2$) to form carbon dioxide ($CO_2$) and water ($H_2O$), according to an\n",
    "equation of the form:\n",
    "\n",
    "$$ (x_1)C_3H_8 + (x_2)O_2 \\longrightarrow (x_3)CO_2 + (x_4)H_2O.$$\n",
    "\n",
    "To “balance” this equation, a chemist must find whole numbers $x_1, \\ldots,  x_4$ such that the\n",
    "total numbers of carbon (C), hydrogen (H), and oxygen (O) atoms on the left match the\n",
    "corresponding numbers of atoms on the right (because atoms are neither destroyed nor\n",
    "created in the reaction).\n",
    "\n",
    "Show how to solve this problem using the algorithm developed in Problem One. Show all work, including traces\n",
    "of any algorithms you run. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 3\n",
    "\n",
    "A certain steam plant burns two types of coal: anthracite (A) and bituminous (B). For each ton of A burned,\n",
    "the plant produces 27.6 million Btu of heat, 3100 grams (g) of sulfur dioxide, and 250 g of particulate matters\n",
    "(pollutants). For each ton of B burned, the plant produces 30.2 million Btu, 6400 g of sulfur dioxide, and 360 g\n",
    "of particuoate matters.\n",
    "\n",
    "\n",
    "(A) How much heat does the steam plant produce when it burns x1 tons of A and x2 tons of B?\n",
    "\n",
    "(B) Suppose the output of the steam plant is described by a vector that lists the amounts of heat, sulfur dioxide,\n",
    "and particulate matter. Express this output as a linear combination of two vectors, assuming the plant burns\n",
    "$x_1$ tons of A and $x_2$ tons of B.\n",
    "\n",
    "(C) Over a certain time period, the steam plant produced 162 million Btu of heat, 23,610 g of sulfur dioxide,\n",
    "and 1623 g of particulate matter. Determine how many tons of each type of coal the steam plant must have\n",
    "burned.\n",
    "\n",
    "For (C), you should find the solution computationally, using your code from Problem 1. \n",
    "\n",
    "Show all work, including a trace of the algorithm's run. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 4\n",
    "\n",
    "For  A -- C,  compute the matrix-vector product *by hand*. Show the dot products for each entry in the result.  If the product is undefined (can not be computed), explain why.\n",
    "\n",
    "(A)  $$\\begin{bmatrix}\n",
    "            1 \\\\\n",
    "            3 \\\\\n",
    "            -2 \\\\\n",
    "          \\end{bmatrix}  @\n",
    "          \\begin{bmatrix}\n",
    "            10 \\\\\n",
    "            -2 \\\\\n",
    "          \\end{bmatrix}$$\n",
    "          \n",
    "          \n",
    "(B)  $$\\begin{bmatrix}\n",
    "            2 & 8 \\\\\n",
    "            -4 & 1 \\\\\n",
    "            3 & 7\n",
    "          \\end{bmatrix}   @\n",
    "          \\begin{bmatrix}\n",
    "            2 \\\\\n",
    "            -1 \\\\\n",
    "          \\end{bmatrix}$$\n",
    "          \n",
    "          \n",
    "(C)  $$\\begin{bmatrix}\n",
    "            8&3&-4 \\\\\n",
    "            5 & 1 & 2\\\\\n",
    "          \\end{bmatrix}   @\n",
    "          \\begin{bmatrix}\n",
    "            1 \\\\\n",
    "            -1 \\\\\n",
    "            2 \\\\\n",
    "          \\end{bmatrix}$$\n",
    "          \n",
    "For D and E, compute the matrix multiplication of the two matrices *by hand.* Show the dot products for each entry in the result. \n",
    "\n",
    "(D)  $$\\begin{bmatrix}\n",
    "            1&3&2 \\\\\n",
    "            0 & -1 & -2\\\\\n",
    "            3 & 2 & 2\\\\\n",
    "          \\end{bmatrix}   @\n",
    "          \\begin{bmatrix}\n",
    "            1 & 2\\\\\n",
    "            -1 & 1\\\\\n",
    "            2 & 3\\\\\n",
    "          \\end{bmatrix}$$\n",
    "          \n",
    "          \n",
    "(E)  $$\\begin{bmatrix}\n",
    "            1&3&1 \\\\\n",
    "            5 & 1 & 1\\\\\n",
    "          \\end{bmatrix}   @\n",
    "          \\begin{bmatrix}\n",
    "            2 & 3& 0 \\\\\n",
    "            -1 & -2 & 3\\\\\n",
    "            1 & -1 & 3\\\\\n",
    "          \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5\n",
    "\n",
    "For each of the parts of Problems 4 which are not undefined, confirm your hand computation by (i) creating `numpy` arrays for each matrix or vector, following the syntax shown in the `numpy` tutorial posted on the class web site, and (ii) perform the given operation, confirming your hand computation. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solution for B\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 6\n",
    "\n",
    "(A) Rewrite the following set of equations as a vector equation: \n",
    "  \n",
    "$$\\begin{aligned}\n",
    "        7x_1 - 3x_2 + 9x_3 &= 21             \\\\       \n",
    "        x_1 + 9x_3         &= 2    \\\\\n",
    "        2x_2 - 4x_3        &= -7               \\\\ \n",
    "  \\end{aligned}$$\n",
    "  \n",
    "(B) Rewrite the set of equations into a matrix equation of the form $A\\mathbf{x} = \\mathbf{b}$, for an appropriate matrix $A$ and vectors $\\bf x$ and $\\bf b$. \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-07-15%20at%2012.03.44%20PM.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 7\n",
    "\n",
    "Given $A$ and $\\bf b$ as below, write the augmented matrix for the linear system that corresponds to the matrix equation $A\\mathbf{x} = \\mathbf{b}$. Then solve the system for $\\bf x$ and write the solution as a vector. (You may do it by hand or using the algorithm, but show your work, e.g., a trace of the run.)\n",
    "\n",
    "![Screen%20Shot%202021-07-15%20at%2012.03.44%20PM.png](attachment:Screen%20Shot%202021-07-15%20at%2012.03.44%20PM.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 8\n",
    "\n",
    "Let ${\\bf u} = \\begin{bmatrix} 2 \\\\ -3 \\\\ 2 \\\\ \\end{bmatrix}$ and $A = \\begin{bmatrix} 5 & 8 & 7 \\\\  0 & 1 & -1 \\\\ \n",
    " 1 & 3 & 0 \\\\ \\end{bmatrix}$. Is $\\bf u$ in the subset of $\\mathbb{R}^3$ spanned by the columns of $A$? Explain why or why not carefully. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**  \n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-06-06%20at%2011.09.20%20AM.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 9\n",
    "\n",
    "Consider the following matrix:\n",
    "\n",
    "![Screen%20Shot%202021-06-06%20at%2011.09.20%20AM.png](attachment:Screen%20Shot%202021-06-06%20at%2011.09.20%20AM.png)\n",
    "\n",
    "(A) Do the columns of $B$ span all of $\\mathbb{R}^4$? Explain carefully why or why not. \n",
    "\n",
    "(B) Does the equation $B\\mathbf{x} = \\mathbf{y}$ have a solution for *every* $y \\in \\mathbb{R}^4$? Explain carefully why or why not. \n",
    "\n",
    "(C) Do the columns of $B$ span all of $\\mathbb{R}^3$? Explain carefully why or why not. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-07-15%20at%2011.18.25%20AM.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 10\n",
    "\n",
    "For the matrix B below, show that the columns span all of $\\mathbb{R}^4$.  Use your GE algorithm from Problem 1 to answer, and show all work, including an execution trace. \n",
    "\n",
    "![Screen%20Shot%202021-07-15%20at%2011.18.25%20AM.png](attachment:Screen%20Shot%202021-07-15%20at%2011.18.25%20AM.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 11\n",
    "\n",
    "\n",
    "Consider the following set of homogeneous equations:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "        x_1  + 2x_2  - 3x_3  &= 0            \\\\       \n",
    "        2x_1 + 5x_2  + 2x_3   &= 0            \\\\ \n",
    "        3x_1 - x_2 - 4x_3  &= 0            \\\\ \n",
    "  \\end{aligned}$$\n",
    "  \n",
    "\n",
    "Describe the solution set to this set of equations. Show all work, including traces of your running the GE algorithm. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 12\n",
    "\n",
    "Consider the following set of homogeneous equations:\n",
    "\n",
    "$$\\begin{aligned}\n",
    "        x_1  + 2x_2  - 3x_3  + 2x_4 - 4 x_5 &= 0            \\\\       \n",
    "        2x_1 + 4x_2  - 5x_3  + x_4  - 6 x_5 &= 0            \\\\ \n",
    "        5x_1 + 10x_2 - 13x_3 + 4x_4 - 16 x_5 &= 0            \\\\ \n",
    "  \\end{aligned}$$\n",
    "  \n",
    "\n",
    "(A) Calculate the *parametric vector form* of the solution to this set of equations, and\n",
    "\n",
    "(B) Give the number of dimensions of the solution set, and the basis for the solution. \n",
    "\n",
    "Show all work, including traces of your running the GE algorithm. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-07-15%20at%2011.20.54%20AM.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 13\n",
    "\n",
    "Consider\n",
    "\n",
    "![Screen%20Shot%202021-07-15%20at%2011.20.54%20AM.png](attachment:Screen%20Shot%202021-07-15%20at%2011.20.54%20AM.png)\n",
    "\n",
    "(A) For what values of $h$ is ${\\bf v}_3\\in \\text{Span}({\\bf v}_1,{\\bf v}_2)$?  Show all work. \n",
    "\n",
    "(B) Let $A$ be the $3\\times 3$ matrix consisting of the columns ${\\bf v}_1,{\\bf v}_2,{\\bf v}_3$ for a given\n",
    "value of $h$.  \n",
    "\n",
    "\n",
    "For what values of $h$ does $A{\\bf x} = {\\bf 0}$ have non-trivial solutions?\n",
    "\n",
    "Show all work. \n",
    "\n",
    "(Hint: Perform GE by hand, carrying the variable $h$ through all the steps; the RREF will answer your questions!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}