fixes error in 2.4

2.10 The Trace Operator/2.10 The Trace Operator.ipynb

@@ -289,9 +289,9 @@
    "source": [
     "Since the transposition of a matrix doesn't change the diagonal, the trace of the matrix is equal to the trace of its transpose:\n",
     "\n",
-    "$\n",
+    "$$\n",
     "\\Tr(\\bs{A})=\\Tr(\\bs{A}^T)\n",
-    "$"
+    "$$"
    ]
   },
   {
@@ -300,9 +300,9 @@
    "source": [
     "## Trace of a product\n",
     "\n",
-    "$\n",
+    "$$\n",
     "\\Tr(\\bs{ABC}) = \\Tr(\\bs{CAB}) = \\Tr(\\bs{BCA})\n",
-    "$\n",
+    "$$\n",
     "\n",
     "\n",
     "### Example 1.\n",

2.4 Linear Dependence and Span/2.4 Linear Dependence and Span.ipynb

@@ -1062,7 +1062,61 @@
     "1. The system must have at least one solution\n",
     "2. Then, the system must have **only** one solution\n",
     "\n",
-    "### Requirement 1. Overdetermined system: the system must have at least one solution for each value of $\\bs{b}$: $m\\geq n$\n",
+    "### Requirement 1. Underdetermined system: the system must have at least one solution for each value of $\\bs{b}$: $n\\geq m$\n",
+    "\n",
+    "<span class='pquote'>\n",
+    "     An underdetermined system of equations is a system with less equations than unknowns\n",
+    "</span>\n",
+    "\n",
+    "If we want our system to have one and only one solution a first requirement is that $n$ must not be bigger than $m$.\n",
+    "\n",
+    "Let's take the example of a ($2\\times 3$) matrix that corresponds to a set of 2 equations with 3 unknowns variables:\n",
+    "\n",
+    "\n",
+    "<div>\n",
+    "$$\n",
+    "\\begin{cases}\n",
+    "8x+y+z=1\\\\\\\\\n",
+    "x+y+z=1\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "</div>\n",
+    "\n",
+    "<div>\n",
+    "$$\n",
+    "x\n",
+    "\\begin{bmatrix}\n",
+    "    8 \\\\\\\\\n",
+    "    1\n",
+    "\\end{bmatrix}\n",
+    "+\n",
+    "y\n",
+    "\\begin{bmatrix}\n",
+    "    1 \\\\\\\\\n",
+    "    1\n",
+    "\\end{bmatrix}\n",
+    "+\n",
+    "z\n",
+    "\\begin{bmatrix}\n",
+    "    1 \\\\\\\\\n",
+    "    1\n",
+    "\\end{bmatrix}\n",
+    "=\n",
+    "\\begin{bmatrix}\n",
+    "    1 \\\\\\\\\n",
+    "    1\n",
+    "\\end{bmatrix}\n",
+    "$$\n",
+    "</div>\n",
+    "\n",
+    "Here is the representation of the planes plotted with the help of this [website](https://technology.cpm.org/general/3dgraph/):\n",
+    "\n",
+    "<img src=\"images/intersection-2-planes-line.png\" alt=\"Plot showing two planes. The intersection of the two planes is a line\" title=\"The intersection of the two planes is a line\" width=\"500\">\n",
+    "<em>The intersection of the two planes is a line</em>\n",
+    "\n",
+    "We can see that in the best case the two planes are not parallel and there are solutions to the set of equations. It means that it exists some points that rely on both planes. But we can also see that there is inevitably an infinite number of points on the intersection (a line that we can see on the figure). We need a third plane to have a unique solution.\n",
+    "\n",
+    "### Requirement 2. Overdetermined system: the system must have **only** one solution for each value of $\\bs{b}$: $m\\geq n$\n",
     "\n",
     "\n",
     "<span class='pquote'>\n",
@@ -1071,6 +1125,7 @@
     "\n",
     "The column figure is helpful to understand why the linear system has usually no solution if $n$ (the number of unknowns) is smaller than $m$ (the number of equations). Let's add 1 equation to the above system in order to end up with a ($3\\times2$) matrix (3 equations and 2 unknowns):\n",
     "\n",
+    "<div>\n",
     "$$\n",
     "\\begin{cases}\n",
     "y = \\frac{1}{2}x+1\\\\\\\\\n",
@@ -1084,9 +1139,11 @@
     "7x-y=2\n",
     "\\end{cases}\n",
     "$$\n",
+    "</div>\n",
     "\n",
     "This corresponds to:\n",
     "\n",
+    "<div>\n",
     "$$\n",
     "x\n",
     "\\begin{bmatrix}\n",
@@ -1108,6 +1165,7 @@
     "    2\n",
     "\\end{bmatrix}\n",
     "$$\n",
+    "</div>\n",
     "\n",
     "So we are still traveling in our 2-dimensional space (see the plot of the column space above) but the point that we are looking for is defined by 3 dimensions. There are cases where the third coordinate does not rely on our 2-dimensional $x$-$y$ plane. In that case no solution exists.\n",
     "\n",
@@ -1253,77 +1311,18 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We would like to go to $b$ but the only path we can take is the blue/orange line. The second equation doesn't provide us with a new direction to take since it is just a linear combination of the first one."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Requirement 2. Underdetermined system: the system must have **only** one solution for each value of $\\bs{b}$: $n=m$\n",
+    "We would like to go to $b$ but the only path we can take is the blue/orange line. The second equation doesn't provide us with a new direction to take since it is just a linear combination of the first one.\n",
     "\n",
-    "<span class='pquote'>\n",
-    "     An underdetermined system of equations is a system with less equations than unknowns\n",
-    "</span>\n",
-    "\n",
-    "We saw that a requirement is that $n$ (the number of unknowns) must not be inferior to $m$ (the number of equations). But if we want our system to have one and only one solution a second requirement is that $n$ must not be bigger than $m$."
+    "Thus, an overdetermined system of independant equations has at most 1 solution."
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Let's take the example of a ($2\\times 3$) matrix that corresponds to a set of 2 equations with 3 unknowns variables:\n",
-    "\n",
-    "\n",
-    "$$\n",
-    "\\begin{cases}\n",
-    "8x+y+z=1\\\\\\\\\n",
-    "x+y+z=1\n",
-    "\\end{cases}\n",
-    "$$\n",
+    "### Square matrix\n",
     "\n",
-    "$$\n",
-    "x\n",
-    "\\begin{bmatrix}\n",
-    "    8 \\\\\\\\\n",
-    "    1\n",
-    "\\end{bmatrix}\n",
-    "+\n",
-    "y\n",
-    "\\begin{bmatrix}\n",
-    "    1 \\\\\\\\\n",
-    "    1\n",
-    "\\end{bmatrix}\n",
-    "+\n",
-    "z\n",
-    "\\begin{bmatrix}\n",
-    "    1 \\\\\\\\\n",
-    "    1\n",
-    "\\end{bmatrix}\n",
-    "=\n",
-    "\\begin{bmatrix}\n",
-    "    1 \\\\\\\\\n",
-    "    1\n",
-    "\\end{bmatrix}\n",
-    "$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Here is the representation of the planes plotted with the help of this [website](https://technology.cpm.org/general/3dgraph/):\n",
-    "\n",
-    "<img src=\"images/intersection-2-planes-line.png\" alt=\"Plot showing two planes. The intersection of the two planes is a line\" title=\"The intersection of the two planes is a line\" width=\"500\">\n",
-    "<em>The intersection of the two planes is a line</em>"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We can see that in the best case the two planes are not parallel and there are solutions to the set of equations. It means that it exists some points that rely on both planes. But we can also see that there is inevitably an infinite number of points on the intersection (a line that we can see on the figure). We need a third plane to have a unique solution.\n",
+    "How could we satisfy both requirements ($m\\geq n$ and $n\\geq m$): we must have $m=n$!\n",
     "\n",
     "The resulting of all of this is that the system needs a **square matrix** $\\bs{A}$ ($m=n$) with linearly independant columns to have a unique solution for every values of $\\bs{b}$.\n",
     "\n",

2.5 Norms/2.5 Norms.ipynb

@@ -157,13 +157,13 @@
     "\n",
     "3- Norms respect the triangle inequity. See bellow.\n",
     "\n",
-    "4- $\\norm{\\bs{k}\\cdot \\bs{u}}=\\norm{\\bs{k}}\\cdot\\norm{\\bs{u}}$. The norm of a vector multiplied by a scalar is equal to the absolute value of this scalar multiplied by the norm of the vector.\n",
+    "4- $\\norm{\\bs{k}\\cdot \\bs{u}}=| \\bs{k}| \\cdot \\norm{\\bs{u}}$. The norm of a vector multiplied by a scalar is equal to the absolute value of this scalar multiplied by the norm of the vector.\n",
     "\n",
     "It is usually written with two horizontal bars: $\\norm{\\bs{x}}$\n",
     "\n",
     "# The triangle inequity\n",
     "\n",
-    "The norm of the sum of some vectors is less than or equal the sum of the norms of these vectors.\n",
+    "The norm of the sum of some vectors is less than or equal to the sum of the norms of these vectors.\n",
     "\n",
     "$$\n",
     "\\norm{\\bs{u}+\\bs{v}} \\leq \\norm{\\bs{u}}+\\norm{\\bs{v}}\n",

2.6 Special Kinds of Matrices and Vectors/2.6 Special Kinds of Matrices and Vectors.ipynb

@@ -991,11 +991,15 @@
    "source": [
     "Let's check that\n",
     "\n",
-    "$\n",
+    "$$\n",
     "\\bs{A^\\text{T}A}=\\bs{AA^\\text{T}}=\\bs{I}\n",
-    "$ and thus $\n",
+    "$$\n",
+    "\n",
+    "and thus\n",
+    "\n",
+    "$$\n",
     "\\bs{A}^\\text{T}=\\bs{A}^{-1}\n",
-    "$"
+    "$$"
    ]
   },
   {

2.7 Eigendecomposition/2.7 Eigendecomposition.ipynb

@@ -783,13 +783,17 @@
     "\n",
     "As we saw it with numpy, if $\\bs{v}$ is an eigenvector of $\\bs{A}$, then any rescaled vector $s\\bs{v}$ is also an eigenvector of $\\bs{A}$. The eigenvalue of the rescaled vector is the same.\n",
     "\n",
-    "Let's try to rescale $\n",
+    "Let's try to rescale\n",
+    "\n",
+    "$$\n",
     "\\bs{v}=\n",
     "\\begin{bmatrix}\n",
     "    1\\\\\\\\\n",
     "    -3\n",
     "\\end{bmatrix}\n",
-    "$ from our preceding example. \n",
+    "$$\n",
+    "\n",
+    "from our preceding example. \n",
     "\n",
     "For instance,\n",
     "\n",

2.8 Singular Value Decomposition/2.8 Singular Value Decomposition.ipynb

@@ -2,12 +2,20 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 3,
    "metadata": {
-    "collapsed": true,
     "scrolled": false
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/Users/lsp/.virtualenvs/kaggle/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.\n",
+      "  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')\n"
+     ]
+    }
+   ],
    "source": [
     "import numpy as np\n",
     "import matplotlib.pyplot as plt\n",
@@ -16,7 +24,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
@@ -39,7 +47,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
@@ -114,7 +122,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
@@ -580,7 +588,7 @@
     "\n",
     "Note that these examples used **diagonal matrices** (all zeros except the diagonal). The general rule is that the transformation associated with diagonal matrices imply only a rescaling of each coordinate **without rotation**. This is a first element to understand the SVD. Look again at the decomposition\n",
     "\n",
-    "<img src=\"images/singularValues.png\" width=\"300\">\n",
+    "<img src=\"images/singular-value-decomposition.png\" width=\"300\">\n",
     "\n",
     "<span class='pquote'>\n",
     "    The transformation associated with diagonal matrices imply only a rescaling of each coordinate **without rotation**\n",
@@ -1945,6 +1953,52 @@
     "\n",
     "- [Frank Cleary - SVD of an image](https://www.frankcleary.com/svdimage/)"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(0, 10)"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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+QqGOzYM6CSLZ8hIKaTcP6iSIZM9LKKTZPKiTIJIPPyeFFJsHdRJE8uHnpFDj5kGdBJH8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\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10a061ed0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "u = [0,0,1,6]\n",
+    "v = [0,0,4,2]\n",
+    "u_bis = [u[2], u[3], v[2],v[3]]\n",
+    "w = [u[0]+v[0], u[1]+v[1], u[2]+v[2], u[3]+v[3]]\n",
+    "\n",
+    "plotVectors([u, u_bis, w], ['red', 'green', 'blue'])\n",
+    "plt.xlim(0, 10)\n",
+    "plt.ylim(0, 10)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {