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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Linear Algebra (Review/Introduction)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Representation of linear equations:\n",
+    "\n",
+    "1. $5x_{1}+3x_{2} =1$\n",
+    "\n",
+    "2. $x_{1}+2x_{2}+3x_{3} =2$\n",
+    "\n",
+    "3. $x_{1}+x_{2}+x_{3} =3$\n",
+    "\n",
+    "in matrix form:\n",
+    "\n",
+    "$\\left[ \\begin{array}{ccc}\n",
+    "5 & 3 & 0 \\\\\n",
+    "1 & 2 & 3 \\\\\n",
+    "1 & 1 & 1 \\end{array} \\right]\n",
+    "\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "x_{3}\\end{array}\\right]=\\left[\\begin{array}{c} \n",
+    "1 \\\\\n",
+    "2 \\\\\n",
+    "3\\end{array}\\right]$\n",
+    "\n",
+    "$Ax=b$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Vectors "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "column vector x (length of 3):\n",
+    "\n",
+    "$\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "x_{3}\\end{array}\\right]$\n",
+    "\n",
+    "row vector y (length of 3):\n",
+    "\n",
+    "$\\left[\\begin{array}{ccc} \n",
+    "y_{1}~ \n",
+    "y_{2}~ \n",
+    "y_{3}\\end{array}\\right]$\n",
+    "\n",
+    "vector of length N:\n",
+    "\n",
+    "$\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "x_{N}\\end{array}\\right]$\n",
+    "\n",
+    "The $i^{th}$ element of x is $x_{i}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Matrices"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "elements in the matrix are denoted $B_{row~column}$\n",
+    "\n",
+    "In general, matrix, B, can be any size, $M \\times N$ (M-rows and N-columns):\n",
+    "\n",
+    "$B=\\left[ \\begin{array}{cccc}\n",
+    "B_{11} & B_{12} &...& B_{1N} \\\\\n",
+    "B_{21} & B_{22} &...& B_{2N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "B_{M1} & B_{M2} &...& B_{MN}\\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Multiplication"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "A column vector is a $1\\times N$ matrix and a row vector is a $M\\times 1$ matrix\n",
+    "\n",
+    "**Multiplying matrices is not commutative**\n",
+    "\n",
+    "$A B \\neq B A$\n",
+    "\n",
+    "Inner dimensions must agree, output is outer dimensions. \n",
+    "\n",
+    "A is $M1 \\times N1$ and B is $M2 \\times N2$\n",
+    "\n",
+    "C=AB\n",
+    "\n",
+    "Therefore N1=M2 and C is $M1 \\times N2$\n",
+    "\n",
+    "If $C'=BA$, then N2=M1 and C is $M2 \\times N1$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "e.g. \n",
+    "$A=\\left[ \\begin{array}{cc}\n",
+    "5 & 3 & 0 \\\\\n",
+    "1 & 2 & 3  \\end{array} \\right]$\n",
+    "\n",
+    "$B=\\left[ \\begin{array}{cccc}\n",
+    "1 & 2 & 3 & 4 \\\\\n",
+    "5 & 6 & 7 & 8 \\\\\n",
+    "9 & 10 & 11 & 12 \\end{array} \\right]$\n",
+    "\n",
+    "C=AB\n",
+    "\n",
+    "$[2\\times 4] = [2 \\times 3][3 \\times 4]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "The rule for multiplying matrices, A and B, is\n",
+    "\n",
+    "$C_{ij} = \\sum_{k=1}^{n}A_{ik}B_{kj}$\n",
+    "\n",
+    "In the previous example, \n",
+    "\n",
+    "$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$\n",
+    "\n",
+    "\n",
+    "Multiplication is associative:\n",
+    "\n",
+    "$(AB)C = A(BC)$\n",
+    "\n",
+    "and distributive:\n",
+    "\n",
+    "$A(B+C)=AB+AC$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   5   3   0\n",
+      "   1   2   3\n",
+      "\n",
+      "B =\n",
+      "\n",
+      "    1    2    3    4\n",
+      "    5    6    7    8\n",
+      "    9   10   11   12\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=[5,3,0;1,2,3] \n",
+    "B=[1,2,3,4;5,6,7,8;9,10,11,12]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "C =\n",
+      "\n",
+      "   20   28    0    0\n",
+      "    0    0    0    0\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "C=zeros(2,4);\n",
+    "C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1);\n",
+    "C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2);\n",
+    "C\n",
+    "%C=A*B"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3)\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "Cp=B*A"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Representation of a problem with Matrices and Vectors"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "You have:\n",
+    "\n",
+    "- a set of known outputs, $y_{1},~y_{2},~...y_{N}$ \n",
+    "\n",
+    "- set of equations that relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## = vector-matrix format:\n",
+    "\n",
+    "$y=Ax=\\left[\\begin{array}{c} \n",
+    "y_{1} \\\\ \n",
+    "y_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "y_{N}\\end{array}\\right]\n",
+    "=\n",
+    "A\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "x_{N}\\end{array}\\right]$\n",
+    "\n",
+    "$A=\n",
+    "\\left[\\begin{array}{cccc} \n",
+    "| & | &  & | \\\\ \n",
+    "a_{1} & a_{2} & ... & a_{N} \\\\ \n",
+    "| & | &  & | \\end{array}\\right]$\n",
+    "\n",
+    "or "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$\n",
+    "\n",
+    "- $a_{i}$ is a column vector \n",
+    "\n",
+    "- $x_{i}$ is a scalar taken from the $i^{th}$ element of x."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Example \n",
+    "\n",
+    "Consider 4 masses connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n",
+    "\n",
+    "![Springs-masses](mass_springs.svg)\n",
+    "\n",
+    "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n",
+    "\n",
+    "$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n",
+    "\n",
+    "$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n",
+    "\n",
+    "$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n",
+    "\n",
+    "$m_{4}g-k(x_{4}-x_{3})=0$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "in matrix form:\n",
+    "\n",
+    "$\\left[ \\begin{array}{cccc}\n",
+    "2k & -k & 0 & 0 \\\\\n",
+    "-k & 2k & -k & 0 \\\\\n",
+    "0 & -k & 2k & -k \\\\\n",
+    "0 & 0 & -k & k \\end{array} \\right]\n",
+    "\\left[ \\begin{array}{c}\n",
+    "x_{1} \\\\\n",
+    "x_{2} \\\\\n",
+    "x_{3} \\\\\n",
+    "x_{4} \\end{array} \\right]=\n",
+    "\\left[ \\begin{array}{c}\n",
+    "m_{1}g \\\\\n",
+    "m_{2}g \\\\\n",
+    "m_{3}g \\\\\n",
+    "m_{4}g \\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "K =\n",
+      "\n",
+      "   20  -10    0    0\n",
+      "  -10   20  -10    0\n",
+      "    0  -10   20  -10\n",
+      "    0    0  -10   10\n",
+      "\n",
+      "y =\n",
+      "\n",
+      "    9.8100\n",
+      "   19.6200\n",
+      "   29.4300\n",
+      "   39.2400\n",
+      "\n",
+      "x =\n",
+      "\n",
+      "    9.8100\n",
+      "   18.6390\n",
+      "   25.5060\n",
+      "   29.4300\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "k=10; % N/m\n",
+    "m1=1; % kg\n",
+    "m2=2;\n",
+    "m3=3;\n",
+    "m4=4;\n",
+    "g=9.81; % m/s^2\n",
+    "K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]\n",
+    "y=[m1*g;m2*g;m3*g;m4*g]\n",
+    "\n",
+    "x=K\\y"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Matrix Operations\n",
+    "\n",
+    "Identity matrix `eye(M,N)` **Assume M=N unless specfied**\n",
+    "\n",
+    "$I_{ij} =1$ if $i=j$ \n",
+    "\n",
+    "$I_{ij} =0$ if $i\\neq j$ \n",
+    "\n",
+    "AI=A=IA\n",
+    "\n",
+    "Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any\n",
+    "number\n",
+    "\n",
+    "$D_{ij} =d_{i}$ if $i=j$ \n",
+    "\n",
+    "$D_{ij} =0$ if $i\\neq j$ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Transpose\n",
+    "\n",
+    "The transpose of a matrix changes the rows -> columns and columns-> rows\n",
+    "\n",
+    "$(A^{T})_{ij} = A_{ji}$\n",
+    "\n",
+    "for diagonal matrices, $D^{T}=D$\n",
+    "\n",
+    "in general, the transpose has the following qualities:\n",
+    "  \n",
+    "1. $(A^{T})^{T}=A$\n",
+    "\n",
+    "2. $(AB)^{T} = B^{T}A^{T}$\n",
+    "\n",
+    "3. $(A+B)^{T} = A^{T} +B^{T}$ \n",
+    "\n",
+    "All matrices have a symmetric and antisymmetric part:\n",
+    "\n",
+    "$A= 1/2(A+A^{T}) +1/2(A-A^{T})$\n",
+    "\n",
+    "If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "   20   38\n",
+      "   28   44\n",
+      "   36   50\n",
+      "   44   56\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   20   38\n",
+      "   28   44\n",
+      "   36   50\n",
+      "   44   56\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   1   1\n",
+      "   1   1\n",
+      "   1   1\n",
+      "   1   1\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "(A*B)'\n",
+    "\n",
+    "B'*A' % if this wasnt true, then inner dimensions wouldn't match\n",
+    "\n",
+    "B'*A' == (A*B)' % 1=True, 0=False"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Square matrix properties\n",
+    "\n",
+    "1. Trace\n",
+    "\n",
+    "2. Inverse\n",
+    "\n",
+    "3. Orthogonal"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Trace\n",
+    "\n",
+    "The trace of a square matrix is the sum of the diagonal elements\n",
+    "\n",
+    "$tr~A=\\sum_{i=1}^{N}A_{ii}$\n",
+    "\n",
+    "The trace is invariant, meaning that if you change the basis through a rotation, then the\n",
+    "trace remains the same. \n",
+    "\n",
+    "It also has the following properties:\n",
+    "\n",
+    "1. $tr~A=tr~A^{T}$\n",
+    "\n",
+    "2. $tr~A+tr~B=tr(A+B)$\n",
+    "\n",
+    "3. $tr(kA)=k tr~A$\n",
+    "\n",
+    "4. $tr(AB)=tr(BA)$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "id_m =\n",
+      "\n",
+      "Diagonal Matrix\n",
+      "\n",
+      "   1   0   0\n",
+      "   0   1   0\n",
+      "   0   0   1\n",
+      "\n",
+      "ans =  3\n"
+     ]
+    }
+   ],
+   "source": [
+    "id_m=eye(3)\n",
+    "trace(id_m)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Inverse\n",
+    "\n",
+    "The inverse of a square matrix, $A^{-1}$ is defined such that\n",
+    "\n",
+    "$A^{-1}A=I=AA^{-1}$\n",
+    "\n",
+    "Not all square matrices have an inverse, they can be *singular* or *non-invertible*\n",
+    "\n",
+    "The inverse has the following properties:\n",
+    "\n",
+    "1. $(A^{-1})^{-1}=A$\n",
+    "\n",
+    "2. $(AB)^{-1}=B^{-1}A^{-1}$\n",
+    "\n",
+    "3. $(A^{-1})^{T}=(A^{T})^{-1}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   0.5762106   0.3533174   0.7172134\n",
+      "   0.7061664   0.4863733   0.9423436\n",
+      "   0.4255961   0.0016613   0.3561407\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=rand(3,3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Ainv =\n",
+      "\n",
+      "   41.5613  -30.1783   -3.8467\n",
+      "   36.2130  -24.2201   -8.8415\n",
+      "  -49.8356   36.1767    7.4460\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Ainv=inv(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "B =\n",
+      "\n",
+      "   0.524529   0.470856   0.708116\n",
+      "   0.084491   0.730986   0.528292\n",
+      "   0.303545   0.782522   0.389282\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "B=rand(3,3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "  -182.185   125.738    40.598\n",
+      "  -133.512    97.116    17.079\n",
+      "   282.422  -200.333   -46.861\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "  -182.185   125.738    40.598\n",
+      "  -133.512    97.116    17.079\n",
+      "   282.422  -200.333   -46.861\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   41.5613   36.2130  -49.8356\n",
+      "  -30.1783  -24.2201   36.1767\n",
+      "   -3.8467   -8.8415    7.4460\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   41.5613   36.2130  -49.8356\n",
+      "  -30.1783  -24.2201   36.1767\n",
+      "   -3.8467   -8.8415    7.4460\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "inv(A*B)\n",
+    "inv(B)*inv(A)\n",
+    "\n",
+    "inv(A')\n",
+    "\n",
+    "inv(A)'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Orthogonal Matrices\n",
+    "\n",
+    "Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A\n",
+    "square matrix is *orthogonal* if all its column vectors are orthogonal to each other and\n",
+    "normalized. The column vectors are then called *orthonormal* and the following results\n",
+    "\n",
+    "$U^{T}U=I=UU^{T}$\n",
+    "\n",
+    "and \n",
+    "\n",
+    "$||Ux||_{2}=||x||_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Determinant\n",
+    "\n",
+    "The **determinant** of a matrix has 3 properties\n",
+    "\n",
+    "1\\. The determinant of the identity matrix is one, $|I|=1$\n",
+    "\n",
+    "2\\. If you multiply a single row by scalar $t$ then the determinant is $t|A|$:\n",
+    "\n",
+    "$t|A|=\\left[ \\begin{array}{cccc}\n",
+    "tA_{11} & tA_{12} &...& tA_{1N} \\\\\n",
+    "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## Determinant (con'd)\n",
+    "3\\. If you switch 2 rows, the determinant changes sign:\n",
+    "$-|A|=\\left[ \\begin{array}{cccc}\n",
+    "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+    "A_{11} & A_{12} &...& A_{1N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## Determinant (con'd)\n",
+    "4\\. inverse of the determinant is the determinant of the inverse:\n",
+    "\n",
+    "$|A^{-1}|=\\frac{1}{|A|}=|A|^{-1}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## Calculating the Determinant\n",
+    "For a $2\\times2$ matrix, \n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{cc}\n",
+    "A_{11} & A_{12} \\\\\n",
+    "A_{21} & A_{22} \\\\\n",
+    "\\end{array} \\right]\\right| = A_{11}A_{22}-A_{21}A_{12}$\n",
+    "\n",
+    "For a $3\\times3$ matrix,\n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{ccc}\n",
+    "A_{11} & A_{12} & A_{13} \\\\\n",
+    "A_{21} & A_{22} & A_{23} \\\\\n",
+    "A_{31} & A_{32} & A_{33}\\end{array} \\right]\\right|=$\n",
+    "\n",
+    "$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}\n",
+    "-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "For larger matrices, the determinant is more involved"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Special Case for determinants\n",
+    "\n",
+    "The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$. \n",
+    "\n",
+    "Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j<i$), the\n",
+    "determinant is \n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{cccc}\n",
+    "A_{11} & A_{12} &...& A_{1N} \\\\\n",
+    "0 & A_{22} &...& A_{2N} \\\\\n",
+    "0 & 0 &\\ddots & \\vdots \\\\\n",
+    "0 & 0 &...& A_{NN}\\end{array} \\right]\\right|=A_{11}A_{22}A_{33}...A_{NN}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "D =\n",
+      "\n",
+      "   1   0   0   0\n",
+      "   0   2   0   0\n",
+      "   0   0   3   0\n",
+      "   0   0   0   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   1   2   3   4\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "D=[1,0,0,0;0,2,0,0;0,0,3,0;0,0,0,4]\n",
+    "A=[1,2,3,4;0,2,3,4;0,0,3,4;0,0,0,4] % upper triangular matrix (4x4)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  24\n",
+      "ans =  24\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(D)\n",
+    "det(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   2   4   6   8\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A(1,:)=2*A(1,:)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  48\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  0.020833\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(inv(A))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  0.020833\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "1/48"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "r1 =\n",
+      "\n",
+      "   2   4   6   8\n",
+      "\n",
+      "r2 =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "   2   4   6   8\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "r1=A(1,:)\n",
+    "r2=A(2,:)\n",
+    "A(1,:)=r2\n",
+    "A(2,:)=r1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans = -48\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(A)"
+   ]
+  }
+ ],
+ "metadata": {
+  "celltoolbar": "Slideshow",
+  "kernelspec": {
+   "display_name": "Octave",
+   "language": "octave",
+   "name": "octave"
+  },
+  "language_info": {
+   "file_extension": ".m",
+   "help_links": [
+    {
+     "text": "MetaKernel Magics",
+     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+    }
+   ],
+   "mimetype": "text/x-octave",
+   "name": "octave",
+   "version": "0.19.14"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/09_Linear-Algebra/.ipynb_checkpoints/lecture_09-checkpoint.ipynb b/09_Linear-Algebra/.ipynb_checkpoints/lecture_09-checkpoint.ipynb
new file mode 100644
index 0000000..482b64e
--- /dev/null
+++ b/09_Linear-Algebra/.ipynb_checkpoints/lecture_09-checkpoint.ipynb
@@ -0,0 +1,1347 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Linear Algebra (Review/Introduction)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Representation of linear equations:\n",
+    "\n",
+    "1. $5x_{1}+3x_{2} =1$\n",
+    "\n",
+    "2. $x_{1}+2x_{2}+3x_{3} =2$\n",
+    "\n",
+    "3. $x_{1}+x_{2}+x_{3} =3$\n",
+    "\n",
+    "in matrix form:\n",
+    "\n",
+    "$\\left[ \\begin{array}{ccc}\n",
+    "5 & 3 & 0 \\\\\n",
+    "1 & 2 & 3 \\\\\n",
+    "1 & 1 & 1 \\end{array} \\right]\n",
+    "\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "x_{3}\\end{array}\\right]=\\left[\\begin{array}{c} \n",
+    "1 \\\\\n",
+    "2 \\\\\n",
+    "3\\end{array}\\right]$\n",
+    "\n",
+    "$Ax=b$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Vectors "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "column vector x (length of 3):\n",
+    "\n",
+    "$\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "x_{3}\\end{array}\\right]$\n",
+    "\n",
+    "row vector y (length of 3):\n",
+    "\n",
+    "$\\left[\\begin{array}{ccc} \n",
+    "y_{1}~ \n",
+    "y_{2}~ \n",
+    "y_{3}\\end{array}\\right]$\n",
+    "\n",
+    "vector of length N:\n",
+    "\n",
+    "$\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "x_{N}\\end{array}\\right]$\n",
+    "\n",
+    "The $i^{th}$ element of x is $x_{i}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "x =\n",
+      "\n",
+      "    1    2    3    4    5    6    7    8    9   10\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "x=[1:10] % row vector"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "    1\n",
+      "    2\n",
+      "    3\n",
+      "    4\n",
+      "    5\n",
+      "    6\n",
+      "    7\n",
+      "    8\n",
+      "    9\n",
+      "   10\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "x' % column vector"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Matrices"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Matrix A is 3x3:\n",
+    "\n",
+    "$A=\\left[ \\begin{array}{ccc}\n",
+    "5 & 3 & 0 \\\\\n",
+    "1 & 2 & 3 \\\\\n",
+    "1 & 1 & 1 \\end{array} \\right]$\n",
+    "\n",
+    "elements in the matrix are denoted $A_{row~column}$, $A_{23}=3$\n",
+    "\n",
+    "In general, matrix, B, can be any size, $M \\times N$ (M-rows and N-columns):\n",
+    "\n",
+    "$B=\\left[ \\begin{array}{cccc}\n",
+    "B_{11} & B_{12} &...& B_{1N} \\\\\n",
+    "B_{21} & B_{22} &...& B_{2N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "B_{M1} & B_{M2} &...& B_{MN}\\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   5   3   0\n",
+      "   1   2   3\n",
+      "   1   1   1\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=[5,3,0;1,2,3;1,1,1]"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Multiplication"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "A column vector is a $1\\times N$ matrix and a row vector is a $M\\times 1$ matrix\n",
+    "\n",
+    "**Multiplying matrices is not commutative**\n",
+    "\n",
+    "$A B \\neq B A$\n",
+    "\n",
+    "Inner dimensions must agree, output is outer dimensions. \n",
+    "\n",
+    "A is $M1 \\times N1$ and B is $M2 \\times N2$\n",
+    "\n",
+    "C=AB\n",
+    "\n",
+    "Therefore N1=M2 and C is $M1 \\times N2$\n",
+    "\n",
+    "If $C'=BA$, then N2=M1 and C is $M2 \\times N1$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "e.g. \n",
+    "$A=\\left[ \\begin{array}{cc}\n",
+    "5 & 3 & 0 \\\\\n",
+    "1 & 2 & 3  \\end{array} \\right]$\n",
+    "\n",
+    "$B=\\left[ \\begin{array}{cccc}\n",
+    "1 & 2 & 3 & 4 \\\\\n",
+    "5 & 6 & 7 & 8 \\\\\n",
+    "9 & 10 & 11 & 12 \\end{array} \\right]$\n",
+    "\n",
+    "C=AB\n",
+    "\n",
+    "$[2\\times 4] = [2 \\times 3][3 \\times 4]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "The rule for multiplying matrices, A and B, is\n",
+    "\n",
+    "$C_{ij} = \\sum_{k=1}^{n}A_{ik}B_{kj}$\n",
+    "\n",
+    "In the previous example, \n",
+    "\n",
+    "$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$\n",
+    "\n",
+    "\n",
+    "Multiplication is associative:\n",
+    "\n",
+    "$(AB)C = A(BC)$\n",
+    "\n",
+    "and distributive:\n",
+    "\n",
+    "$A(B+C)=AB+AC$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   5   3   0\n",
+      "   1   2   3\n",
+      "\n",
+      "B =\n",
+      "\n",
+      "    1    2    3    4\n",
+      "    5    6    7    8\n",
+      "    9   10   11   12\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=[5,3,0;1,2,3] \n",
+    "B=[1,2,3,4;5,6,7,8;9,10,11,12]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "C =\n",
+      "\n",
+      "   20   28    0    0\n",
+      "    0    0    0    0\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "C=zeros(2,4);\n",
+    "C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1);\n",
+    "C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2);\n",
+    "C\n",
+    "%C=A*B"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3)\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "Cp=B*A"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Representation of a problem with Matrices and Vectors"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "You have:\n",
+    "\n",
+    "- a set of known outputs, $y_{1},~y_{2},~...y_{N}$ \n",
+    "\n",
+    "- set of equations that relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## = vector-matrix format:\n",
+    "\n",
+    "$y=Ax=\\left[\\begin{array}{c} \n",
+    "y_{1} \\\\ \n",
+    "y_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "y_{N}\\end{array}\\right]\n",
+    "=\n",
+    "A\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "x_{N}\\end{array}\\right]$\n",
+    "\n",
+    "$A=\n",
+    "\\left[\\begin{array}{cccc} \n",
+    "| & | &  & | \\\\ \n",
+    "a_{1} & a_{2} & ... & a_{N} \\\\ \n",
+    "| & | &  & | \\end{array}\\right]$\n",
+    "\n",
+    "or "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$\n",
+    "\n",
+    "- $a_{i}$ is a column vector \n",
+    "\n",
+    "- $x_{i}$ is a scalar taken from the $i^{th}$ element of x."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Example \n",
+    "\n",
+    "Consider 4 masses connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n",
+    "\n",
+    "![Springs-masses](mass_springs.svg)\n",
+    "\n",
+    "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n",
+    "\n",
+    "$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n",
+    "\n",
+    "$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n",
+    "\n",
+    "$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n",
+    "\n",
+    "$m_{4}g-k(x_{4}-x_{3})=0$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "in matrix form:\n",
+    "\n",
+    "$\\left[ \\begin{array}{cccc}\n",
+    "2k & -k & 0 & 0 \\\\\n",
+    "-k & 2k & -k & 0 \\\\\n",
+    "0 & -k & 2k & -k \\\\\n",
+    "0 & 0 & -k & k \\end{array} \\right]\n",
+    "\\left[ \\begin{array}{c}\n",
+    "x_{1} \\\\\n",
+    "x_{2} \\\\\n",
+    "x_{3} \\\\\n",
+    "x_{4} \\end{array} \\right]=\n",
+    "\\left[ \\begin{array}{c}\n",
+    "m_{1}g \\\\\n",
+    "m_{2}g \\\\\n",
+    "m_{3}g \\\\\n",
+    "m_{4}g \\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "K =\n",
+      "\n",
+      "   20  -10    0    0\n",
+      "  -10   20  -10    0\n",
+      "    0  -10   20  -10\n",
+      "    0    0  -10   10\n",
+      "\n",
+      "y =\n",
+      "\n",
+      "    9.8100\n",
+      "   19.6200\n",
+      "   29.4300\n",
+      "   39.2400\n",
+      "\n",
+      "x =\n",
+      "\n",
+      "    9.8100\n",
+      "   18.6390\n",
+      "   25.5060\n",
+      "   29.4300\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "k=10; % N/m\n",
+    "m1=1; % kg\n",
+    "m2=2;\n",
+    "m3=3;\n",
+    "m4=4;\n",
+    "g=9.81; % m/s^2\n",
+    "K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]\n",
+    "y=[m1*g;m2*g;m3*g;m4*g]\n",
+    "\n",
+    "x=K\\y"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Matrix Operations\n",
+    "\n",
+    "Identity matrix `eye(M,N)` **Assume M=N unless specfied**\n",
+    "\n",
+    "$I_{ij} =1$ if $i=j$ \n",
+    "\n",
+    "$I_{ij} =0$ if $i\\neq j$ \n",
+    "\n",
+    "AI=A=IA\n",
+    "\n",
+    "Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any\n",
+    "number\n",
+    "\n",
+    "$D_{ij} =d_{i}$ if $i=j$ \n",
+    "\n",
+    "$D_{ij} =0$ if $i\\neq j$ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Transpose\n",
+    "\n",
+    "The transpose of a matrix changes the rows -> columns and columns-> rows\n",
+    "\n",
+    "$(A^{T})_{ij} = A_{ji}$\n",
+    "\n",
+    "for diagonal matrices, $D^{T}=D$\n",
+    "\n",
+    "in general, the transpose has the following qualities:\n",
+    "  \n",
+    "1. $(A^{T})^{T}=A$\n",
+    "\n",
+    "2. $(AB)^{T} = B^{T}A^{T}$\n",
+    "\n",
+    "3. $(A+B)^{T} = A^{T} +B^{T}$ \n",
+    "\n",
+    "All matrices have a symmetric and antisymmetric part:\n",
+    "\n",
+    "$A= 1/2(A+A^{T}) +1/2(A-A^{T})$\n",
+    "\n",
+    "If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "   20   38\n",
+      "   28   44\n",
+      "   36   50\n",
+      "   44   56\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   20   38\n",
+      "   28   44\n",
+      "   36   50\n",
+      "   44   56\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   1   1\n",
+      "   1   1\n",
+      "   1   1\n",
+      "   1   1\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "(A*B)'\n",
+    "\n",
+    "B'*A' % if this wasnt true, then inner dimensions wouldn't match\n",
+    "\n",
+    "B'*A' == (A*B)' % 1=True, 0=False"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Square matrix properties\n",
+    "\n",
+    "1. Trace\n",
+    "\n",
+    "2. Inverse\n",
+    "\n",
+    "3. Orthogonal"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Trace\n",
+    "\n",
+    "The trace of a square matrix is the sum of the diagonal elements\n",
+    "\n",
+    "$tr~A=\\sum_{i=1}^{N}A_{ii}$\n",
+    "\n",
+    "The trace is invariant, meaning that if you change the basis through a rotation, then the\n",
+    "trace remains the same. \n",
+    "\n",
+    "It also has the following properties:\n",
+    "\n",
+    "1. $tr~A=tr~A^{T}$\n",
+    "\n",
+    "2. $tr~A+tr~B=tr(A+B)$\n",
+    "\n",
+    "3. $tr(kA)=k tr~A$\n",
+    "\n",
+    "4. $tr(AB)=tr(BA)$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "id_m =\n",
+      "\n",
+      "Diagonal Matrix\n",
+      "\n",
+      "   1   0   0\n",
+      "   0   1   0\n",
+      "   0   0   1\n",
+      "\n",
+      "ans =  3\n"
+     ]
+    }
+   ],
+   "source": [
+    "id_m=eye(3)\n",
+    "trace(id_m)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Inverse\n",
+    "\n",
+    "The inverse of a square matrix, $A^{-1}$ is defined such that\n",
+    "\n",
+    "$A^{-1}A=I=AA^{-1}$\n",
+    "\n",
+    "Not all square matrices have an inverse, they can be *singular* or *non-invertible*\n",
+    "\n",
+    "The inverse has the following properties:\n",
+    "\n",
+    "1. $(A^{-1})^{-1}=A$\n",
+    "\n",
+    "2. $(AB)^{-1}=B^{-1}A^{-1}$\n",
+    "\n",
+    "3. $(A^{-1})^{T}=(A^{T})^{-1}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   0.5762106   0.3533174   0.7172134\n",
+      "   0.7061664   0.4863733   0.9423436\n",
+      "   0.4255961   0.0016613   0.3561407\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=rand(3,3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Ainv =\n",
+      "\n",
+      "   41.5613  -30.1783   -3.8467\n",
+      "   36.2130  -24.2201   -8.8415\n",
+      "  -49.8356   36.1767    7.4460\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Ainv=inv(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "B =\n",
+      "\n",
+      "   0.524529   0.470856   0.708116\n",
+      "   0.084491   0.730986   0.528292\n",
+      "   0.303545   0.782522   0.389282\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "B=rand(3,3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "  -182.185   125.738    40.598\n",
+      "  -133.512    97.116    17.079\n",
+      "   282.422  -200.333   -46.861\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "  -182.185   125.738    40.598\n",
+      "  -133.512    97.116    17.079\n",
+      "   282.422  -200.333   -46.861\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   41.5613   36.2130  -49.8356\n",
+      "  -30.1783  -24.2201   36.1767\n",
+      "   -3.8467   -8.8415    7.4460\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   41.5613   36.2130  -49.8356\n",
+      "  -30.1783  -24.2201   36.1767\n",
+      "   -3.8467   -8.8415    7.4460\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "inv(A*B)\n",
+    "inv(B)*inv(A)\n",
+    "\n",
+    "inv(A')\n",
+    "\n",
+    "inv(A)'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Orthogonal Matrices\n",
+    "\n",
+    "Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A\n",
+    "square matrix is *orthogonal* if all its column vectors are orthogonal to each other and\n",
+    "normalized. The column vectors are then called *orthonormal* and the following results\n",
+    "\n",
+    "$U^{T}U=I=UU^{T}$\n",
+    "\n",
+    "and \n",
+    "\n",
+    "$||Ux||_{2}=||x||_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Determinant\n",
+    "\n",
+    "The **determinant** of a matrix has 3 properties\n",
+    "\n",
+    "1. The determinant of the identity matrix is one, $|I|=1$\n",
+    "\n",
+    "2. If you multiply a single row by scalar $t$ then the determinant is $t|A|$:\n",
+    "\n",
+    "$t|A|=\\left[ \\begin{array}{cccc}\n",
+    "tA_{11} & tA_{12} &...& tA_{1N} \\\\\n",
+    "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n",
+    "\n",
+    "3. If you switch 2 rows, the determinant changes sign:\n",
+    "\n",
+    "\n",
+    "$-|A|=\\left[ \\begin{array}{cccc}\n",
+    "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+    "A_{11} & A_{12} &...& A_{1N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n",
+    "\n",
+    "4. inverse of the determinant is the determinant of the inverse:\n",
+    "\n",
+    "$|A^{-1}|=\\frac{1}{|A|}=|A|^{-1}$\n",
+    "\n",
+    "For a $2\\times2$ matrix, \n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{cc}\n",
+    "A_{11} & A_{12} \\\\\n",
+    "A_{21} & A_{22} \\\\\n",
+    "\\end{array} \\right]\\right| = A_{11}A_{22}-A_{21}A_{12}$\n",
+    "\n",
+    "For a $3\\times3$ matrix,\n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{ccc}\n",
+    "A_{11} & A_{12} & A_{13} \\\\\n",
+    "A_{21} & A_{22} & A_{23} \\\\\n",
+    "A_{31} & A_{32} & A_{33}\\end{array} \\right]\\right|=$\n",
+    "\n",
+    "$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}\n",
+    "-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$\n",
+    "\n",
+    "For larger matrices, the determinant is \n",
+    "\n",
+    "$|A|=\n",
+    "\\frac{1}{n!}\n",
+    "\\sum_{i_{r}}^{N}\n",
+    "\\sum_{j_{r}}^{N}\n",
+    "\\epsilon_{i_{1}...i_{N}}\n",
+    "\\epsilon_{j_{1}...j_{N}}\n",
+    "A_{i_{1}j_{1}}\n",
+    "A_{i_{2}j_{2}}\n",
+    "...\n",
+    "A_{i_{N}j_{N}}$\n",
+    "\n",
+    "Where the Levi-Cevita symbol is $\\epsilon_{i_{1}i_{2}...i_{N}} = 1$ if it is an even\n",
+    "permutation and $\\epsilon_{i_{1}i_{2}...i_{N}} = -1$ if it is an odd permutation and\n",
+    "$\\epsilon_{i_{1}i_{2}...i_{N}} = 0$ if $i_{n}=i_{m}$ e.g. $i_{1}=i_{7}$\n",
+    "\n",
+    "![Levi-Cevita rule for even and odd permutations for an $N\\times N$\n",
+    "determinant](even_odd_perm.svg)\n",
+    "\n",
+    "So a $4\\times4$ matrix determinant,\n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{cccc}\n",
+    "A_{11} & A_{12} & A_{13} & A_{14} \\\\\n",
+    "A_{21} & A_{22} & A_{23} & A_{24} \\\\\n",
+    "A_{31} & A_{32} & A_{33} & A_{34} \\\\\n",
+    "A_{41} & A_{42} & A_{43} & A_{44}  \\end{array} \\right]\\right|$\n",
+    "\n",
+    "$=\\epsilon_{1234}\\epsilon_{1234}A_{11}A_{22}A_{33}A_{44}+\n",
+    "\\epsilon_{2341}\\epsilon_{1234}A_{21}A_{32}A_{43}A_{14}+\n",
+    "\\epsilon_{3412}\\epsilon_{1234}A_{31}A_{42}A_{13}A_{24}+\n",
+    "\\epsilon_{4123}\\epsilon_{1234}A_{41}A_{12}A_{23}A_{34}+...\n",
+    "\\epsilon_{4321}\\epsilon_{4321}A_{44}A_{33}A_{22}A_{11}$\n",
+    "\n",
+    "### Special Case for determinants\n",
+    "\n",
+    "The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$. \n",
+    "\n",
+    "Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j<i$), the\n",
+    "determinant is \n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{cccc}\n",
+    "A_{11} & A_{12} &...& A_{1N} \\\\\n",
+    "0 & A_{22} &...& A_{2N} \\\\\n",
+    "0 & 0 &\\ddots & \\vdots \\\\\n",
+    "0 & 0 &...& A_{NN}\\end{array} \\right]\\right|=A_{11}A_{22}A_{33}...A_{NN}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "D =\n",
+      "\n",
+      "   1   0   0   0\n",
+      "   0   2   0   0\n",
+      "   0   0   3   0\n",
+      "   0   0   0   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   1   2   3   4\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "D=[1,0,0,0;0,2,0,0;0,0,3,0;0,0,0,4]\n",
+    "A=[1,2,3,4;0,2,3,4;0,0,3,4;0,0,0,4] % upper triangular matrix (4x4)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  24\n",
+      "ans =  24\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(D)\n",
+    "det(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   2   4   6   8\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A(1,:)=2*A(1,:)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  48\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  0.020833\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(inv(A))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  0.020833\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "1/48"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "r1 =\n",
+      "\n",
+      "   2   4   6   8\n",
+      "\n",
+      "r2 =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "   2   4   6   8\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "r1=A(1,:)\n",
+    "r2=A(2,:)\n",
+    "A(1,:)=r2\n",
+    "A(2,:)=r1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans = -48\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(A)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Working with matrices (or arrays)\n",
+    "\n",
+    "When you need a row of your matrix, use `A(1,:)`\n",
+    "\n",
+    "When you need a column of your matrix, use `A(:,1)`"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 42,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "B =\n",
+      "\n",
+      " Columns 1 through 6:\n",
+      "\n",
+      "   9.3678e-02   6.6083e-01   8.6163e-01   2.4581e-01   3.2483e-01   6.9693e-01\n",
+      "   4.5401e-01   5.6867e-01   1.4516e-01   9.5440e-02   6.8525e-01   6.1453e-03\n",
+      "   5.4096e-01   9.6443e-01   8.2454e-02   1.8644e-01   7.0369e-01   2.9403e-03\n",
+      "   8.3509e-01   1.1006e-01   5.8880e-01   4.9634e-01   9.3148e-02   8.3110e-01\n",
+      "   5.9499e-01   8.8513e-01   6.5640e-01   5.3244e-01   2.9278e-01   7.6347e-02\n",
+      "   1.5456e-01   3.1275e-01   4.5873e-01   7.2297e-01   3.6996e-01   1.2478e-01\n",
+      "   4.7960e-01   1.8491e-01   6.1816e-01   3.3450e-01   5.9307e-01   8.9796e-01\n",
+      "   9.2872e-01   3.8687e-01   8.3991e-01   8.8064e-01   6.0058e-01   3.5300e-01\n",
+      "   7.3063e-01   2.0543e-01   7.0693e-01   5.0973e-01   7.1647e-01   1.3979e-01\n",
+      "   2.3365e-01   3.5988e-01   4.8453e-01   9.6969e-01   9.8916e-01   2.7608e-02\n",
+      "\n",
+      " Columns 7 and 8:\n",
+      "\n",
+      "   8.8934e-01   1.7184e-01\n",
+      "   2.5948e-03   7.2273e-01\n",
+      "   1.1076e-01   9.3209e-01\n",
+      "   8.3082e-01   3.6414e-01\n",
+      "   9.6406e-01   2.4431e-01\n",
+      "   2.6633e-04   9.9584e-01\n",
+      "   4.4733e-01   4.2158e-01\n",
+      "   1.2252e-01   8.7513e-01\n",
+      "   5.2620e-01   8.7209e-01\n",
+      "   5.8721e-02   9.7262e-01\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "B=rand(10,8)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 44,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "   0.861626\n",
+      "   0.145156\n",
+      "   0.082454\n",
+      "   0.588799\n",
+      "   0.656403\n",
+      "   0.458729\n",
+      "   0.618158\n",
+      "   0.839911\n",
+      "   0.706933\n",
+      "   0.484532\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "B(:,3) % print 3 column of matrix B"
+   ]
+  }
+ ],
+ "metadata": {
+  "celltoolbar": "Slideshow",
+  "kernelspec": {
+   "display_name": "Octave",
+   "language": "octave",
+   "name": "octave"
+  },
+  "language_info": {
+   "file_extension": ".m",
+   "help_links": [
+    {
+     "text": "MetaKernel Magics",
+     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+    }
+   ],
+   "mimetype": "text/x-octave",
+   "name": "octave",
+   "version": "0.19.14"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/09_Linear-Algebra/.lecture_09.md.swp b/09_Linear-Algebra/.lecture_09.md.swp
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diff --git a/09_Linear-Algebra/09_Linear-Algebra.ipynb b/09_Linear-Algebra/09_Linear-Algebra.ipynb
new file mode 100644
index 0000000..71dc415
--- /dev/null
+++ b/09_Linear-Algebra/09_Linear-Algebra.ipynb
@@ -0,0 +1,1173 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Linear Algebra (Review/Introduction)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "Representation of linear equations:\n",
+    "\n",
+    "1. $5x_{1}+3x_{2} =1$\n",
+    "\n",
+    "2. $x_{1}+2x_{2}+3x_{3} =2$\n",
+    "\n",
+    "3. $x_{1}+x_{2}+x_{3} =3$\n",
+    "\n",
+    "in matrix form:\n",
+    "\n",
+    "$\\left[ \\begin{array}{ccc}\n",
+    "5 & 3 & 0 \\\\\n",
+    "1 & 2 & 3 \\\\\n",
+    "1 & 1 & 1 \\end{array} \\right]\n",
+    "\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "x_{3}\\end{array}\\right]=\\left[\\begin{array}{c} \n",
+    "1 \\\\\n",
+    "2 \\\\\n",
+    "3\\end{array}\\right]$\n",
+    "\n",
+    "$Ax=b$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Vectors "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "column vector x (length of 3):\n",
+    "\n",
+    "$\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "x_{3}\\end{array}\\right]$\n",
+    "\n",
+    "row vector y (length of 3):\n",
+    "\n",
+    "$\\left[\\begin{array}{ccc} \n",
+    "y_{1}~ \n",
+    "y_{2}~ \n",
+    "y_{3}\\end{array}\\right]$\n",
+    "\n",
+    "vector of length N:\n",
+    "\n",
+    "$\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "x_{N}\\end{array}\\right]$\n",
+    "\n",
+    "The $i^{th}$ element of x is $x_{i}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Matrices"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "elements in the matrix are denoted $B_{row~column}$\n",
+    "\n",
+    "In general, matrix, B, can be any size, $M \\times N$ (M-rows and N-columns):\n",
+    "\n",
+    "$B=\\left[ \\begin{array}{cccc}\n",
+    "B_{11} & B_{12} &...& B_{1N} \\\\\n",
+    "B_{21} & B_{22} &...& B_{2N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "B_{M1} & B_{M2} &...& B_{MN}\\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Multiplication"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "A column vector is a $1\\times N$ matrix and a row vector is a $M\\times 1$ matrix\n",
+    "\n",
+    "**Multiplying matrices is not commutative**\n",
+    "\n",
+    "$A B \\neq B A$\n",
+    "\n",
+    "Inner dimensions must agree, output is outer dimensions. \n",
+    "\n",
+    "A is $M1 \\times N1$ and B is $M2 \\times N2$\n",
+    "\n",
+    "C=AB\n",
+    "\n",
+    "Therefore N1=M2 and C is $M1 \\times N2$\n",
+    "\n",
+    "If $C'=BA$, then N2=M1 and C is $M2 \\times N1$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "e.g. \n",
+    "$A=\\left[ \\begin{array}{cc}\n",
+    "5 & 3 & 0 \\\\\n",
+    "1 & 2 & 3  \\end{array} \\right]$\n",
+    "\n",
+    "$B=\\left[ \\begin{array}{cccc}\n",
+    "1 & 2 & 3 & 4 \\\\\n",
+    "5 & 6 & 7 & 8 \\\\\n",
+    "9 & 10 & 11 & 12 \\end{array} \\right]$\n",
+    "\n",
+    "C=AB\n",
+    "\n",
+    "$[2\\times 4] = [2 \\times 3][3 \\times 4]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "The rule for multiplying matrices, A and B, is\n",
+    "\n",
+    "$C_{ij} = \\sum_{k=1}^{n}A_{ik}B_{kj}$\n",
+    "\n",
+    "In the previous example, \n",
+    "\n",
+    "$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$\n",
+    "\n",
+    "\n",
+    "Multiplication is associative:\n",
+    "\n",
+    "$(AB)C = A(BC)$\n",
+    "\n",
+    "and distributive:\n",
+    "\n",
+    "$A(B+C)=AB+AC$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   5   3   0\n",
+      "   1   2   3\n",
+      "\n",
+      "B =\n",
+      "\n",
+      "    1    2    3    4\n",
+      "    5    6    7    8\n",
+      "    9   10   11   12\n",
+      "\n",
+      "C =\n",
+      "\n",
+      "   20   28   36   44\n",
+      "   38   44   50   56\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=[5,3,0;1,2,3] \n",
+    "B=[1,2,3,4;5,6,7,8;9,10,11,12]\n",
+    "C=A*B;\n",
+    "C"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "C =\n",
+      "\n",
+      "   20   28    0    0\n",
+      "    0    0    0    0\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "C=zeros(2,4);\n",
+    "C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1);\n",
+    "C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2);\n",
+    "C\n",
+    "%C=A*B"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3)\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "Cp=B*A"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Representation of a problem with Matrices and Vectors"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "You have:\n",
+    "\n",
+    "- a set of known outputs, $y_{1},~y_{2},~...y_{N}$ \n",
+    "\n",
+    "- set of equations that relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## = vector-matrix format:\n",
+    "\n",
+    "$y=Ax=\\left[\\begin{array}{c} \n",
+    "y_{1} \\\\ \n",
+    "y_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "y_{N}\\end{array}\\right]\n",
+    "=\n",
+    "A\\left[\\begin{array}{c} \n",
+    "x_{1} \\\\ \n",
+    "x_{2} \\\\\n",
+    "\\vdots  \\\\\n",
+    "x_{N}\\end{array}\\right]$\n",
+    "\n",
+    "$A=\n",
+    "\\left[\\begin{array}{cccc} \n",
+    "| & | &  & | \\\\ \n",
+    "a_{1} & a_{2} & ... & a_{N} \\\\ \n",
+    "| & | &  & | \\end{array}\\right]$\n",
+    "\n",
+    "or "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$\n",
+    "\n",
+    "- $a_{i}$ is a column vector \n",
+    "\n",
+    "- $x_{i}$ is a scalar taken from the $i^{th}$ element of x."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "# Example \n",
+    "\n",
+    "Consider 4 masses connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n",
+    "\n",
+    "![Springs-masses](mass_springs.svg)\n",
+    "\n",
+    "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n",
+    "\n",
+    "$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n",
+    "\n",
+    "$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n",
+    "\n",
+    "$m_{4}g-k(x_{4}-x_{3})=0$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "in matrix form:\n",
+    "\n",
+    "$\\left[ \\begin{array}{cccc}\n",
+    "2k & -k & 0 & 0 \\\\\n",
+    "-k & 2k & -k & 0 \\\\\n",
+    "0 & -k & 2k & -k \\\\\n",
+    "0 & 0 & -k & k \\end{array} \\right]\n",
+    "\\left[ \\begin{array}{c}\n",
+    "x_{1} \\\\\n",
+    "x_{2} \\\\\n",
+    "x_{3} \\\\\n",
+    "x_{4} \\end{array} \\right]=\n",
+    "\\left[ \\begin{array}{c}\n",
+    "m_{1}g \\\\\n",
+    "m_{2}g \\\\\n",
+    "m_{3}g \\\\\n",
+    "m_{4}g \\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "x =\n",
+      "\n",
+      "    9.8100\n",
+      "   18.6390\n",
+      "   25.5060\n",
+      "   29.4300\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "k=10; % N/m\n",
+    "m1=1; % kg\n",
+    "m2=2;\n",
+    "m3=3;\n",
+    "m4=4;\n",
+    "g=9.81; % m/s^2\n",
+    "K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k];\n",
+    "y=[m1*g;m2*g;m3*g;m4*g];\n",
+    "\n",
+    "x=K\\y"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Special Matrices\n",
+    "\n",
+    "Identity matrix `eye(M,N)` **Assume M=N unless specfied**\n",
+    "\n",
+    "$I_{ij} =1$ if $i=j$ \n",
+    "\n",
+    "$I_{ij} =0$ if $i\\neq j$ \n",
+    "\n",
+    "AI=A=IA\n",
+    "\n",
+    "Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any\n",
+    "number\n",
+    "\n",
+    "$D_{ij} =d_{i}$ if $i=j$ \n",
+    "\n",
+    "$D_{ij} =0$ if $i\\neq j$ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## Matrix operations\n",
+    "### Transpose\n",
+    "\n",
+    "The transpose of a matrix changes the rows -> columns and columns-> rows\n",
+    "\n",
+    "$(A^{T})_{ij} = A_{ji}$\n",
+    "\n",
+    "for diagonal matrices, $D^{T}=D$\n",
+    "\n",
+    "in general, the transpose has the following qualities:\n",
+    "  \n",
+    "1. $(A^{T})^{T}=A$\n",
+    "\n",
+    "2. $(AB)^{T} = B^{T}A^{T}$\n",
+    "\n",
+    "3. $(A+B)^{T} = A^{T} +B^{T}$ \n",
+    "\n",
+    "All matrices have a symmetric and antisymmetric part:\n",
+    "\n",
+    "$A= 1/2(A+A^{T}) +1/2(A-A^{T})$\n",
+    "\n",
+    "If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "   20   38\n",
+      "   28   44\n",
+      "   36   50\n",
+      "   44   56\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   20   38\n",
+      "   28   44\n",
+      "   36   50\n",
+      "   44   56\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "   1   1\n",
+      "   1   1\n",
+      "   1   1\n",
+      "   1   1\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "(A*B)'\n",
+    "\n",
+    "B'*A' % if this wasnt true, then inner dimensions wouldn't match\n",
+    "\n",
+    "B'*A' == (A*B)' % 1=True, 0=False"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "## Square matrix properties\n",
+    "\n",
+    "1. Trace\n",
+    "\n",
+    "2. Inverse\n",
+    "\n",
+    "3. Orthogonal"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "### Trace\n",
+    "\n",
+    "The trace of a square matrix is the sum of the diagonal elements\n",
+    "\n",
+    "$tr~A=\\sum_{i=1}^{N}A_{ii}$\n",
+    "\n",
+    "The trace is invariant, meaning that if you change the basis through a rotation, then the\n",
+    "trace remains the same. \n",
+    "\n",
+    "It also has the following properties:\n",
+    "\n",
+    "1. $tr~A=tr~A^{T}$\n",
+    "\n",
+    "2. $tr~A+tr~B=tr(A+B)$\n",
+    "\n",
+    "3. $tr(kA)=k tr~A$\n",
+    "\n",
+    "4. $tr(AB)=tr(BA)$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "id_m =\n",
+      "\n",
+      "Diagonal Matrix\n",
+      "\n",
+      "   1   0   0\n",
+      "   0   4   0\n",
+      "   0   0   1\n",
+      "\n",
+      "ans =  6\n"
+     ]
+    }
+   ],
+   "source": [
+    "id_m=eye(3);\n",
+    "id_m(2,2)=4\n",
+    "trace(id_m)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Inverse\n",
+    "\n",
+    "The inverse of a square matrix, $A^{-1}$ is defined such that\n",
+    "\n",
+    "$A^{-1}A=I=AA^{-1}$\n",
+    "\n",
+    "Not all square matrices have an inverse, they can be *singular* or *non-invertible*\n",
+    "\n",
+    "The inverse has the following properties:\n",
+    "\n",
+    "1. $(A^{-1})^{-1}=A$\n",
+    "\n",
+    "2. $(AB)^{-1}=B^{-1}A^{-1}$\n",
+    "\n",
+    "3. $(A^{-1})^{T}=(A^{T})^{-1}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   0.71624   0.28713   0.91248\n",
+      "   0.43423   0.50924   0.89731\n",
+      "   0.81776   0.15682   0.48825\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A=rand(3,3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Ainv =\n",
+      "\n",
+      "  -1.189272  -0.032028   2.281478\n",
+      "  -5.750167   4.369462   2.716129\n",
+      "   3.838847  -1.349811  -2.645516\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "Ainv=inv(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "B =\n",
+      "\n",
+      "   0.88372   0.43914   0.13476\n",
+      "   0.43439   0.34519   0.30337\n",
+      "   0.84604   0.86479   0.21558\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "B=rand(3,3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =\n",
+      "\n",
+      "  -1.189272  -5.750167   3.838847\n",
+      "  -0.032028   4.369462  -1.349811\n",
+      "   2.281478   2.716129  -2.645516\n",
+      "\n",
+      "ans =\n",
+      "\n",
+      "  -1.189272  -5.750167   3.838847\n",
+      "  -0.032028   4.369462  -1.349811\n",
+      "   2.281478   2.716129  -2.645516\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "%inv(A*B)\n",
+    "%inv(B)*inv(A)\n",
+    "\n",
+    "inv(A')\n",
+    "\n",
+    "inv(A)'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Orthogonal Matrices\n",
+    "\n",
+    "Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A\n",
+    "square matrix is *orthogonal* if all its column vectors are orthogonal to each other and\n",
+    "normalized. The column vectors are then called *orthonormal* and the following results\n",
+    "\n",
+    "$U^{T}U=I=UU^{T}$\n",
+    "\n",
+    "and \n",
+    "\n",
+    "$||Ux||_{2}=||x||_{2}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Determinant\n",
+    "\n",
+    "The **determinant** of a matrix has 3 properties\n",
+    "\n",
+    "1\\. The determinant of the identity matrix is one, $|I|=1$\n",
+    "\n",
+    "2\\. If you multiply a single row by scalar $t$ then the determinant is $t|A|$:\n",
+    "\n",
+    "$t|A|=\\left[ \\begin{array}{cccc}\n",
+    "tA_{11} & tA_{12} &...& tA_{1N} \\\\\n",
+    "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## Determinant (con'd)\n",
+    "3\\. If you switch 2 rows, the determinant changes sign:\n",
+    "$-|A|=\\left[ \\begin{array}{cccc}\n",
+    "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+    "A_{11} & A_{12} &...& A_{1N} \\\\\n",
+    "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+    "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## Determinant (con'd)\n",
+    "4\\. inverse of the determinant is the determinant of the inverse:\n",
+    "\n",
+    "$|A^{-1}|=\\frac{1}{|A|}=|A|^{-1}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "source": [
+    "## Calculating the Determinant\n",
+    "For a $2\\times2$ matrix, \n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{cc}\n",
+    "A_{11} & A_{12} \\\\\n",
+    "A_{21} & A_{22} \\\\\n",
+    "\\end{array} \\right]\\right| = A_{11}A_{22}-A_{21}A_{12}$\n",
+    "\n",
+    "For a $3\\times3$ matrix,\n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{ccc}\n",
+    "A_{11} & A_{12} & A_{13} \\\\\n",
+    "A_{21} & A_{22} & A_{23} \\\\\n",
+    "A_{31} & A_{32} & A_{33}\\end{array} \\right]\\right|=$\n",
+    "\n",
+    "$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}\n",
+    "-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "source": [
+    "For larger matrices, the determinant is more involved"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "slideshow": {
+     "slide_type": "slide"
+    }
+   },
+   "source": [
+    "### Special Case for determinants\n",
+    "\n",
+    "The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$. \n",
+    "\n",
+    "Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j<i$), the\n",
+    "determinant is \n",
+    "\n",
+    "$|A|=\\left|\\left[ \\begin{array}{cccc}\n",
+    "A_{11} & A_{12} &...& A_{1N} \\\\\n",
+    "0 & A_{22} &...& A_{2N} \\\\\n",
+    "0 & 0 &\\ddots & \\vdots \\\\\n",
+    "0 & 0 &...& A_{NN}\\end{array} \\right]\\right|=A_{11}A_{22}A_{33}...A_{NN}$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "D =\n",
+      "\n",
+      "   1   0   0   0\n",
+      "   0   2   0   0\n",
+      "   0   0   3   0\n",
+      "   0   0   0   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   1   2   3   4\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "D=[1,0,0,0;0,2,0,0;0,0,3,0;0,0,0,4]\n",
+    "A=[1,2,3,4;0,2,3,4;0,0,3,4;0,0,0,4] % upper triangular matrix (4x4)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  24\n",
+      "ans =  24\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(D)\n",
+    "det(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "A =\n",
+      "\n",
+      "   2   4   6   8\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "A(1,:)=2*A(1,:)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  48\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  0.020833\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(inv(A))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans =  0.020833\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "1/48"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "subslide"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "r1 =\n",
+      "\n",
+      "   2   4   6   8\n",
+      "\n",
+      "r2 =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "   0   2   3   4\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n",
+      "A =\n",
+      "\n",
+      "   0   2   3   4\n",
+      "   2   4   6   8\n",
+      "   0   0   3   4\n",
+      "   0   0   0   4\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "r1=A(1,:)\n",
+    "r2=A(2,:)\n",
+    "A(1,:)=r2\n",
+    "A(2,:)=r1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "collapsed": false,
+    "slideshow": {
+     "slide_type": "fragment"
+    }
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "ans = -48\r\n"
+     ]
+    }
+   ],
+   "source": [
+    "det(A)"
+   ]
+  }
+ ],
+ "metadata": {
+  "celltoolbar": "Slideshow",
+  "kernelspec": {
+   "display_name": "Octave",
+   "language": "octave",
+   "name": "octave"
+  },
+  "language_info": {
+   "file_extension": ".m",
+   "help_links": [
+    {
+     "text": "MetaKernel Magics",
+     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+    }
+   ],
+   "mimetype": "text/x-octave",
+   "name": "octave",
+   "version": "0.19.14"
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+ "nbformat_minor": 2
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diff --git a/09_Linear-Algebra/octave-workspace b/09_Linear-Algebra/octave-workspace
new file mode 100644
index 0000000..e69de29