09_Lecture_I_06_Transposes_Permutations_Spaces.ipynb

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    "+ This notebook is part of lecture 5 *Transposes, permutations, and vector spaces* in the OCW MIT course 18.06 by Prof Gilbert Strang [1]\n",
    "+ Created by me, Dr Juan H Klopper\n",
    "    + Head of Acute Care Surgery\n",
    "    + Groote Schuur Hospital\n",
    "    + University Cape Town\n",
    "    + <a href=\"mailto:juan.klopper@uct.ac.za\">Email me with your thoughts, comments, suggestions and corrections</a> \n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\"><img alt=\"Creative Commons Licence\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" href=\"http://purl.org/dc/dcmitype/InteractiveResource\" property=\"dct:title\" rel=\"dct:type\">Linear Algebra OCW MIT18.06</span> <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">IPython notebook [2] study notes by Dr Juan H Klopper</span> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc/4.0/\">Creative Commons Attribution-NonCommercial 4.0 International License</a>.\n",
    "\n",
    "+ [1] <a href=\"http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/index.htm\">OCW MIT 18.06</a>\n",
    "+ [2] Fernando Pérez, Brian E. Granger, IPython: A System for Interactive Scientific Computing, Computing in Science and Engineering, vol. 9, no. 3, pp. 21-29, May/June 2007, doi:10.1109/MCSE.2007.53. URL: http://ipython.org"
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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image('KRG_elegant_logo_for_light_BG.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'Image' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-1-0aa83734b505>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mImage\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m'UCT_logo.gif'\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mNameError\u001b[0m: name 'Image' is not defined"
     ]
    }
   ],
   "source": [
    "Image('UCT_logo.gif')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import init_printing, Matrix, symbols\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Transposes, permutations and vector spaces"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The permutation matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Permutation matrices, usually denoted as $P$, have the property that $P^{-1}=p^T$.  This means that the inverse of the matrix is equal to the transpose of the matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have not discussed inverses in detail, but for now, remember that it is easy to calculate the inverse of a square matrix (one that is invertible), by using the `.inv()` method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Permutation matrices allow for row exchanges as elementary operation.  They are useful when dealing with $0$'s in pivot positions.  Below, we create a matrix, `P`, which is an $n=3$ identity matrix with rows $1$ and $2$ exchanged."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  1  0⎤\n",
       "⎢       ⎥\n",
       "⎢1  0  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  0  1⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = Matrix([[0, 1, 0], [1, 0, 0], [0, 0, 1]])\n",
    "P # Exchanging rows 1 and 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transpose of a matrix interchanges the rows and columns.  We can calculate the transpose using the `,transpose()` method.  Below we call the inverse and transpose methods on the matrix `P`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0 & 1 & 0\\\\1 & 0 & 0\\\\0 & 0 & 1\\end{matrix}\\right]\\right )$$"
      ],
      "text/plain": [
       "⎛⎡0  1  0⎤  ⎡0  1  0⎤⎞\n",
       "⎜⎢       ⎥  ⎢       ⎥⎟\n",
       "⎜⎢1  0  0⎥, ⎢1  0  0⎥⎟\n",
       "⎜⎢       ⎥  ⎢       ⎥⎟\n",
       "⎝⎣0  0  1⎦  ⎣0  0  1⎦⎠"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.inv(), P.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They are indeed equal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.inv() == P.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For a matrix of size $n \\times n$ then there are $n!$ number of permutations.  This is easy to conceptualize.  When starting with an $n=3$ square matrix, there are $n$ rows available for interchanging.  Once one is chosen, there are $n-1$ left, and so on, hence $n!$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The transpose of a matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have briefly mentioned transposes of a matrix, but what are they?  They simply make rows of the column elements and columns of the row elements as in the example below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34 = symbols('a11, a12, a13, a14, a21, a22, a23, a24, a31, a32, a33, a34')\n",
    "# Creating mathematical scalar constants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI8AAABLCAMAAAB+xCQDAAAAOVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACXHtMAAAAEnRSTlMAMquJZkQiEEAwmXbvu91UzXwVE4xGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADUUlEQVRoBe2a4XKjMAyEbaBwh4HkeP+HPduwWAJZMTM3aa5jflRga9FaOFO+Nsau8WjM9x7LZsMYu7adP76+144JHrpmDX76b7aSynfVT2qGcFb7IzSFDH14f/qxdb0jfl+d3hYYXcH78zX5T/68Dq9cHPO3BeaFgvkZps5XGpdQbuiesSpivDj/EARD2zZNfkGConeda+x+a+ZnjJ1ZWj9nnYu2EM9OtmtBMPqZbUWiRFA0vp6b9mzmZ5r96LBuvzge0Y8xiNLtBUFQ9yuWexEJCuuT2+1hGEP9DGvoTBgJB3wgbqPspyTo/LPK+5EU4ZZL2Cjh4H4efmR+mkf4fQYfiCH7dAyrJPCr3Vt7SveXssI9j4809WPCcLeMJrSpxI8sMHbJ72exhBlmGGJ+hsa1tm9cfProC+J1sX65kqCf83ZkRXjF2Hcc88NKwgcim5Qu9sTev9fZsheYqBgmb6Vf9w30z/30s7V2VDpElrKtIDzdbtoVOT99+1xH/1ARyV3kUyRO8a1TzuGjUNjWuQYNzfnh0vddVT96r2t/an/0Duizdf/U/ugd0Gfr/rnVHx3WhFvdFvwQHjywrpQHIUAUehmGCA8is4gHgXXFPAgBYsYP4UFkXnjw1/p7UxNYS1invT8LgiQUHQkKxoN/yN/rKKwlrFP8SIIklPxIipBXwIPAOtUP5UEIUrw64jyIzCIeBNYpfjgPQnDEqx2TUZTwILBO88N4EAJEyQ7nwZT5mgcPrNP80JIQINI5+XzLLOPBhHWFfiBAlC3QUWQW8SCwDtBGbySeQ4AoJrFBZFYeZG3RL+r7T+2P3gF9tu6f2h+9A/ps3T//dX9u491twQ/hwQPSSnkQAlBeZpsQHoQCMSrY54vAGiCtmAchAOVl/AgloLz6IbCWIE17XxUE5TyIEogXPxTWwuQGaYofSVDOg6kESvkR+rw4rAHSVD+UByEA5YV650MukZTMzwnWdkhT/MiCGzwIDkTk/eGwhn/aaX4YD0KQKO/cHX8tKqD08/R5ETmBNM1PUiRBKQ9CgbjdK+PHJEgr83MIQHnJau4MJRBjXs4PIK2YByEA5eVcpHEoEFU/Sfbes1x/3usiVftMP5/1fcghfhPxM74P2XXmL2AiOPBfHPPmAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a_{11} & a_{12} & a_{13}\\\\a_{21} & a_{22} & a_{23}\\\\a_{31} & a_{32} & a_{33}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a₁₁  a₁₂  a₁₃⎤\n",
       "⎢             ⎥\n",
       "⎢a₂₁  a₂₂  a₂₃⎥\n",
       "⎢             ⎥\n",
       "⎣a₃₁  a₃₂  a₃₃⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[a11, a12, a13], [a21, a22, a23], [a31, a32, a33]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a_{11} & a_{21} & a_{31}\\\\a_{12} & a_{22} & a_{32}\\\\a_{13} & a_{23} & a_{33}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a₁₁  a₂₁  a₃₁⎤\n",
       "⎢             ⎥\n",
       "⎢a₁₂  a₂₂  a₃₂⎥\n",
       "⎢             ⎥\n",
       "⎣a₁₃  a₂₃  a₃₃⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note how the elements in the first row became the elements of the first column."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need not stick with square matrices, this applies to any sized matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\\\a_{21} & a_{22} & a_{23} & a_{24}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a₁₁  a₁₂  a₁₃  a₁₄⎤\n",
       "⎢                  ⎥\n",
       "⎣a₂₁  a₂₂  a₂₃  a₂₄⎦"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[a11, a12, a13, a14], [a21, a22, a23, a24]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a_{11} & a_{21}\\\\a_{12} & a_{22}\\\\a_{13} & a_{23}\\\\a_{14} & a_{24}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a₁₁  a₂₁⎤\n",
       "⎢        ⎥\n",
       "⎢a₁₂  a₂₂⎥\n",
       "⎢        ⎥\n",
       "⎢a₁₃  a₂₃⎥\n",
       "⎢        ⎥\n",
       "⎣a₁₄  a₂₄⎦"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This means that a matrix of size $m \\times n$ becomes a matrix of size $n \\times m$ after transposing it."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiplying a matrix by its transpose results in a symmetric matrix (detail in the next section)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a_{11}^{2} + a_{12}^{2} + a_{13}^{2} + a_{14}^{2} & a_{11} a_{21} + a_{12} a_{22} + a_{13} a_{23} + a_{14} a_{24}\\\\a_{11} a_{21} + a_{12} a_{22} + a_{13} a_{23} + a_{14} a_{24} & a_{21}^{2} + a_{22}^{2} + a_{23}^{2} + a_{24}^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡         2      2      2      2                                             ⎤\n",
       "⎢      a₁₁  + a₁₂  + a₁₃  + a₁₄         a₁₁⋅a₂₁ + a₁₂⋅a₂₂ + a₁₃⋅a₂₃ + a₁₄⋅a₂₄⎥\n",
       "⎢                                                                            ⎥\n",
       "⎢                                                2      2      2      2      ⎥\n",
       "⎣a₁₁⋅a₂₁ + a₁₂⋅a₂₂ + a₁₃⋅a₂₃ + a₁₄⋅a₂₄        a₂₁  + a₂₂  + a₂₃  + a₂₄       ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * A.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symmetric matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A symmetric matrix is a square matrix with elements opposite the main diagonal all equal to each other."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is an example with integers as elements.  Note the values across from each other along the main diagonal, i.e. the $29$ and the $29$, the $15$ and the $15$, then the $43,16$ and $16,43$ and so on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}35 & 29 & 15 & 43\\\\29 & 29 & 16 & 42\\\\15 & 16 & 9 & 23\\\\43 & 42 & 23 & 61\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡35  29  15  43⎤\n",
       "⎢              ⎥\n",
       "⎢29  29  16  42⎥\n",
       "⎢              ⎥\n",
       "⎢15  16  9   23⎥\n",
       "⎢              ⎥\n",
       "⎣43  42  23  61⎦"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = Matrix([[1, 5, 3], [3, 4, 2], [2, 2, 1], [4, 6, 3]])\n",
    "S = B * B.transpose()\n",
    "S"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Symmetric matrices are equal to their inverses."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S == S.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector spaces"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A vector space consists of a set of vectors with useful properties.  Think of the common $\\mathbb{R}^2$.  Every point $\\left( a,b \\right) \\in \\mathbb{R}$ is a vector.  We can fill all (reach all of the points) of $\\mathbb{R}$ with an infinite set of vectors.  Note that these vector spaces also include the zero vector $\\mathbf{0}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When considering the familiar vector space $\\mathbb{R}^2$ it is clear to see that we can take a subset of the vectors space and through linear combinations of the, fill all of the space.  Think of the two vectors $\\left(1,0\\right)$ and $\\left(0,1\\right)$  These are the two vectors, each of length $1$ and respectively along the $x$ and $y$ axes.  A constant multiple of each, added together (what we call a linear combination) will fill the vector space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These two vectors in our example are then called _basis vectors_ and we say that they _span_ the vector space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Most texts denote a vector space as $V$.  Basis vectors in $V$ allow us to use scalar multiplication and addition which are then closed operations, that is, the result is still an element of the vector space."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A subspace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A subspace is a subset of a vector space.  It must still allow for the closure property mentioned above.  This means that one of the quadrants of $V=\\mathbb{R}^2$ cannot be a vector subspace.  Scalar multiplication and vector addition can result in a vector that is not in that quadrant."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is important to note that the zero vector is a subspace, trivial though it may be.  In fact, all vector subspaces must contain the zero vector.  The other trivial subspace of a vector space, $V$, is $V$ itself."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another example of a subspace in $\\mathbb{R}^2$ is a line through the origin.  Addition of any vectors on that line or a scalar multiple of any such vectors results in a vector that will still be on that line (closure)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In $\\mathbb{R}^3$, the zero vector, a line through the origin, and a plane through the origin are all vector subspaces."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Column spaces of matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The idea of the column space of a matrix is one that we have seen before.  It is a very important way to consider a matrix.  Here we see each column in a matrix as a vector.  In (1) below, we depict $ A \\mathbf{x}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\begin{bmatrix}a_{11} & a_{12} & a_{13} \\\\ a_{21} & a_{22} & a_{23} \\\\ a_{31} & a_{32} & a_{33} \\end{bmatrix} \\begin{bmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{bmatrix}  \\tag{1}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can rewrite this as in (2)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$x_1 \\begin{bmatrix} a_{11} \\\\ a_{21} \\\\ a_{31} \\end{bmatrix} + x_2 \\begin{bmatrix} a_{12} \\\\ a_{22} \\\\ a_{32} \\end{bmatrix} + x_3 \\begin{bmatrix} a_{13} \\\\ a_{23} \\\\ a_{33} \\end{bmatrix}\\tag{2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This flows naturally when you consider that $A \\mathbf{x}$ as a linear set of three equations written in matrix form, i.e. we started as in (3) and cerated the matrix of coefficients $A$ and the column vector $\\mathbf{x}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 \\\\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 \\\\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 \\tag{3}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Giving this linear system a right-hand side, as in (4), we create another column vector, $\\mathbf{b}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1 \\\\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2 \\\\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = b_3 \\tag{4}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In matrix notation we take (2) and end with (5) below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$x_1 \\begin{bmatrix} a_{11} \\\\ a_{21} \\\\ a_{31} \\end{bmatrix} + x_2 \\begin{bmatrix} a_{12} \\\\ a_{22} \\\\ a_{32} \\end{bmatrix} + x_3 \\begin{bmatrix} a_{13} \\\\ a_{23} \\\\ a_{33} \\end{bmatrix} = \\begin{bmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{bmatrix}\\tag{5}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now see that we put constraints on $\\mathbf{b}$.  If we view the problem as a linear combination of the three column vectors that make up $A$, we have that they are three vectors in a vector space and that $\\mathbf{b}$ must be another vector in that vector space."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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