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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## CS 132 Homework 03 B  \n",
    "\n",
    "### Due Wednesday July 28th 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.  Each problem is worth 10 points. \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": 1,
   "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": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Gaussian Elimination\n",
    "\n",
    "# number of digits of precision to print out\n",
    "\n",
    "prec = 4\n",
    "\n",
    "########################################################################################################\n",
    "\n",
    "# Returns the Row Echelon (upper triangular) form\n",
    "\n",
    "def forwardElimination(A,traceLevel=0):\n",
    "    \n",
    "    A = (np.copy(A)).astype(float)\n",
    "    \n",
    "    if (traceLevel > 0):\n",
    "        print(\"\\nRunning Forward Elimination on:\\n\")\n",
    "        print(np.round(A, decimals=4),\"\\n\")\n",
    "        print()\n",
    "    \n",
    "    (numRows,numCols) = A.shape\n",
    "    \n",
    "    # r = row we are currently working on         pivot value is A[r][c]\n",
    "    r = 0            \n",
    "    for c in range(numCols):     # solve for variable in column c \n",
    "        # find row in column c with first non-zero element, and exchange with row r                  \n",
    "        r1 = r\n",
    "        while(r1 < numRows):\n",
    "            if (not np.isclose(A[r1][c],0.0)):   # A[r1][c] is first non-zero element at or below r in column c\n",
    "                break\n",
    "            r1 += 1\n",
    "        \n",
    "        if(r1 == numRows):    # all zeros below r in this column\n",
    "            continue          # go on to the next column, but still working on same row   \n",
    "         \n",
    "        if(r1 != r): \n",
    "            # exchange rows r1 and r\n",
    "            A[[r1,r],:] = A[[r,r1],:] \n",
    "            if (traceLevel == 2): \n",
    "                print(\"Exchange R\" + str(r1+1) + \" and R\" + str(r+1) + \"\\n\") \n",
    "                print(np.round(A, decimals=4))                \n",
    "                print() \n",
    "\n",
    "        # now use pivot A[r][c] to eliminate all vars in this column below row r\n",
    "        for r2 in range(r+1,numRows):\n",
    "            rowReduce(A,r,c,r2,traceLevel)\n",
    "            \n",
    "        r += 1  \n",
    "        if (r >= numRows):\n",
    "            break\n",
    "            \n",
    "    return A\n",
    "\n",
    "# for pivot A[r][c], eliminate variable in location A[r2][c] in row r2 using row operations\n",
    "\n",
    "def rowReduce(A,r,c,r2,traceLevel=0):\n",
    "\n",
    "    if(not np.isclose(A[r2][c],0.0)):\n",
    "\n",
    "        factor = -A[r2][c]/A[r][c] \n",
    "        A[r2] += factor * A[r]\n",
    "            \n",
    "        if(traceLevel == 2):\n",
    "            print(\"R\" + str(r2+1) + \" += \" + str(np.around(factor,prec)) + \"*R\" + str(r+1) + \"\\n\")  \n",
    "            print(np.round(A, decimals=4))\n",
    "            print()\n",
    "\n",
    "# Take a Row Echelon Form and return a Reduced Row Echelon Form\n",
    "\n",
    "def backwardSubstitute(A,augmented=True,traceLevel=0): \n",
    "    \n",
    "    numRows,numCols = A.shape\n",
    "    \n",
    "    if (A.dtype != 'float64'):\n",
    "        A = A.astype(float)\n",
    "\n",
    "    # now back-substitute the variables from bottom row to top\n",
    "    if (traceLevel > 0):\n",
    "        print(\"Creating Reduced Row Echelon Form...\\n\") \n",
    "\n",
    "    for r in range(numRows):\n",
    "\n",
    "        # find variable in this row\n",
    "        for c in range(numCols):\n",
    "            if(not np.isclose(A[r][c],0.0)):\n",
    "                break \n",
    "       \n",
    "        if ((augmented and c >= numCols-1) or (c >= numCols)):        # inconsistent or redundant row\n",
    "                continue \n",
    "                        \n",
    "        # A[r][c] is variable to eliminate\n",
    "        \n",
    "        factor = A[r][c]\n",
    "        \n",
    "        if (np.isclose(factor,0.0)):   # already eliminated\n",
    "            continue\n",
    "        \n",
    "        if(not np.isclose(factor,1.0)):  \n",
    "            A[r] *= 1/factor\n",
    "            if (traceLevel == 2):\n",
    "                print(\"R\" + str(r+1) + \" = R\" + str(r+1) + \"/\" + str(np.around(factor,prec)) + \"\\n\")  \n",
    "                print(np.round(A, decimals=4))\n",
    "                print()\n",
    "\n",
    "        for r2 in range(r): \n",
    "            rowReduce(A,r,c,r2,traceLevel)\n",
    "            \n",
    "    return A \n",
    "\n",
    "    \n",
    "# try to find row of all zeros except for last column, in augmented matrix. \n",
    "\n",
    "def noSolution(A):\n",
    "    numRows,numCols = A.shape\n",
    "    for r in range(numRows-1,-1,-1):         # start from bottom, since inconsistent rows end up there\n",
    "        for c in range(numCols):\n",
    "            if(not np.isclose(A[r][c],0.0)):  # found first non-0 in this row\n",
    "                if(c == numCols-1):\n",
    "                    return True\n",
    "                else:\n",
    "                    break\n",
    "    return False\n",
    "\n",
    "########################################################################################################\n",
    "\n",
    "# Runs GE and returns a reduced row echelon form\n",
    "\n",
    "# If b == None assumes that A is NOT an augmented matrix, and runs GE and returns Reduced Row Echelon Form\n",
    "\n",
    "# If b is a column matrix then adjoins it to A and runs GE;\n",
    "#       Always returns the Reduced Row Echelon Form\n",
    "#       If inconsistent then also prints out \"Inconsistent!\"\n",
    "\n",
    "# If b is a length n array instead of a 1 x n array (column vector)\n",
    "# b will be converted to a column vector, for convenience. \n",
    "\n",
    "# traceLevel 0 will not print anything out during the run\n",
    "# traceLevel 1 will print out various stages of the process and the intermediate matrices\n",
    "# traceLevel 2 will also print out the row operations used at each step. \n",
    "\n",
    "# If you want to produce an Echelon Form (NOT reduced), then use forwardElimination instead. \n",
    "\n",
    "# See examples for more explanation of how to use this\n",
    "\n",
    "def GaussianElimination(A,b=None, traceLevel = 0):\n",
    "    if( type(b) != type(None)):\n",
    "        if( (A.shape)[0] == 1 ):\n",
    "            b = np.array([b]).T\n",
    "        Ab = (np.hstack((A.copy(),b))).astype(float)\n",
    "    else:\n",
    "        Ab = A.copy().astype(float)\n",
    "        \n",
    "    if( traceLevel > 0 and type(b) == type(None)):\n",
    "        print(\"\\nCreating Reduced Row Echelon Form:\\n\")\n",
    "        print(np.round(Ab, decimals=4))\n",
    "    elif( traceLevel > 0 ):\n",
    "        print(\"\\nRunning Gaussian Elimination on augmented matrix:\\n\")\n",
    "        print(np.round(Ab, decimals=4))\n",
    "    \n",
    "    B = forwardElimination(Ab,traceLevel)\n",
    "    \n",
    "    if( traceLevel > 0 ):\n",
    "        print(\"\\nEchelon Form:\\n\")\n",
    "        print( np.round(B, decimals=4) + np.zeros(B.shape),\"\\n\")       # adding -0.0 + 0 gives 0.0\n",
    "    \n",
    "    if ( type(b) != type(None) and noSolution(B) ):\n",
    "        print(\"No solution!\")\n",
    "\n",
    "    C = backwardSubstitute(B,type(b)!=type(None),traceLevel)\n",
    "        \n",
    "    if( traceLevel > 0 ):\n",
    "        print(\"Reduced Row Echelon Form:\\n\")\n",
    "        print( np.round(C, decimals=4) + np.zeros(C.shape),\"\\n\")   # adding -0.0 + 0 gives 0.0\n",
    "            \n",
    "    return C\n",
    "\n",
    "########################################################################################################\n",
    "  \n"
   ]
  },
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"
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 5\n",
    "\n",
    "(A) Consider the following vectors, the last one of which has an unknown value $h$:\n",
    "\n",
    "\n",
    "![Screen%20Shot%202021-07-23%20at%203.12.04%20PM.png](attachment:Screen%20Shot%202021-07-23%20at%203.12.04%20PM.png)\n",
    "\n",
    "For what values of $h$ is the set of vectors *linearly dependent*?\n",
    "\n",
    "(B) Again, for this set of vectors:\n",
    "\n",
    "![Screen%20Shot%202021-07-23%20at%203.12.31%20PM.png](attachment:Screen%20Shot%202021-07-23%20at%203.12.31%20PM.png)\n",
    "\n",
    "\n",
    "for what values of $h$ is the set of vectors *linearly dependent*?\n",
    "\n",
    "(C)  Consider a $2\\times 2$ matrix with linearly dependent columns. List all the possible echelon forms of this\n",
    "matrix, using $\\blacksquare$ for a pivot, $0$ for a zero, and $\\ast$ for any real value.\n",
    "\n",
    "(D) Consider a $2\\times 3$ matrix whose column space is all of $\\mathbb{R}^2$. List all the possible echelon forms of this\n",
    "matrix, using $\\blacksquare$ for a pivot, $0$ for a zero, and $\\ast$ for any real value.\n",
    "\n",
    "(E) Consider a $4\\times 3$ matrix $A$ whose columns are linearly independent.  List all the possible echelon forms of this matrix, using $\\blacksquare$ for a pivot, $0$ for a zero, and $\\ast$ for any real value.\n",
    "\n",
    "\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "attachments": {
    "Screen%20Shot%202021-07-23%20at%203.38.13%20PM.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 6\n",
    "\n",
    "(A) Find all ${\\bf x}\\in \\mathbb{R}^4$ that are mapped into the zero vector by the transformation ${\\bf x}\\mapsto A{\\bf x}$ for\n",
    "\n",
    "![Screen%20Shot%202021-07-23%20at%203.38.13%20PM.png](attachment:Screen%20Shot%202021-07-23%20at%203.38.13%20PM.png)\n",
    "\n",
    "Show all work, e.g., traces of running the GE algorithm. \n",
    "\n",
    "(B) Let ${\\bf b} = \\begin{bmatrix} -1 \\\\ 3 \\\\ -1 \\\\ 4 \\end{bmatrix}$.  Answer the following question *without* running the GE algorithm on the augmented $A:b$, but by just considering the RREF created in Part A:  **Is ${\\bf b}$ in the span of the columns of $A$?**   Explain carefully why running the GE algorithm again is not necessary to answer the question. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 7\n",
    "\n",
    "Transformation $S$ and $T$ *commute* if for every vector $\\bf x$, \n",
    "\n",
    "$$S(T({\\bf x})) \\ = \\ T(S({\\bf x})).$$\n",
    "\n",
    "(A)  Consider the following geometric 2D transformations: \n",
    "\n",
    "- D -- a dilation (in which x-coordinates and y-coordinates are scaled by the same factor $d$); \n",
    "- R -- a rotation (about the origin, for some $\\theta$), \n",
    "- F -- a reflection across the $x$-axis, \n",
    "- S -- a shear (by some factor $f$ in the $x$-direction), \n",
    "\n",
    "Assume each of these transformations is fixed with specific parameters (e.g., a specific line for F, or some $\\theta$ for R).\n",
    "\n",
    "Which pairs of transformations commute? (There are 6 pairs to consider.)  Justify briefly in each case. \n",
    "\n",
    "(B)  Let us define an *x-dilation* as a 2D transformation which multiplies every $x$ value by a scalar $a$,\n",
    "and a *y-dilation* as a 2D transformation which multiplies every $y$ value by a scalar $b$.  Prove/show the\n",
    "following fact:  For any $a$ and $b$, the corresponding x-dilation and y-dilation commute. \n",
    "\n",
    "(C) From your result in Part B, you may conclude *immediately* that a reflection across the x-axis commutes\n",
    "with a reflection across the y-axis.  Why is this the case?\n",
    "\n",
    "Hint: When the professor in your math class asks you to justify/show/explain why, he/she is NOT asking\n",
    "for intuition expressed in English...."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 8\n",
    "\n",
    "(A)  Find a  matrix that implements the linear transformation\n",
    "\n",
    "\n",
    "$$T\\left (\\begin{bmatrix} x_1 \\\\  x_2 \\\\ \\end{bmatrix} \\right ) = \n",
    "\\begin{bmatrix} 2x_2 - 3x_1 \\\\ x_1 - 4x_2 \\\\ 0 \\\\ x_2\\end{bmatrix} $$\n",
    "\n",
    "\n",
    "(B) Find a vector in $\\mathbb{R}^2$ which is mapped to $\\begin{bmatrix}-29\\\\ -7\\\\ 0\\\\ 5\\end{bmatrix}$ by the transformation in (A). \n",
    "\n",
    "\n",
    "(C) Is $T$ *onto*? Explain why or why not.\n",
    "\n",
    "(D) Is $T$ *1-to-1*? Explain why or why not. \n",
    "\n",
    "Hint: For C and D, think about the pivots...."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 8 \n",
    "\n",
    "(A) Find a matrix that implements the linear transformation \n",
    "\n",
    "$$T'\\left (\\begin{bmatrix} x_1 \\\\  x_2 \\\\ x_3 \\\\  x_4 \\\\ \\end{bmatrix} \\right ) = \n",
    "\\begin{bmatrix} 2x_1 + 3x_3 - 4x_4 \\end{bmatrix} $$\n",
    "\n",
    "(B) Find a vector in $\\mathbb{R}^4$ which is mapped to $\\begin{bmatrix}5\\end{bmatrix}$ by the transformation in (A). \n",
    "\n",
    "(C) Is $T$ *onto*? Explain why or why not.\n",
    "\n",
    "(D) Is $T$ *1-to-1*? Explain why or why not. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem 9\n",
    "\n",
    "(A) Suppose $(B-C)D = {\\bf 0}$, where $B$ and $C$ are $m\\times n$ matrices and $D$ is invertible. Show that $B =C$.\n",
    "\n",
    "(B)  Suppose $A$ and $B$ are $n\\times n$, $B$ is invertible, and $AB$ is invertible. Show that $A$ is invertible. (Hint: Let\n",
    "$C = AB$, and solve the equation for $A$.)\n",
    "\n",
    "(C) Suppose $P$ is invertible and $A = PBP^{-1}$. Solve for $B$ in terms of $A$.\n",
    "\n",
    "(D) Suppose that $AB = I_n$ and $BC = I_n$. Show that $B=C$. "
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Solution:**\n",
    "\n"
   ]
  }
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