From 14101abcf9b0e9e79af7b6ec53c5cee5f4dc47b4 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Aur=C3=A9lien=20Geron?= <ageron@users.noreply.github.com>
Date: Thu, 20 Apr 2017 21:45:58 +0200
Subject: [PATCH] Fix typo s/Algeabra/Algebra/

---
 math_linear_algebra.ipynb | 1287 +++++++++++++++++++++++++++++--------
 1 file changed, 1002 insertions(+), 285 deletions(-)

diff --git a/math_linear_algebra.ipynb b/math_linear_algebra.ipynb
index bf28a30..df6e95a 100644
--- a/math_linear_algebra.ipynb
+++ b/math_linear_algebra.ipynb
@@ -2,7 +2,10 @@
  "cells": [
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "**Math - Linear Algebra**\n",
     "\n",
@@ -13,7 +16,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Before we start, let's ensure that this notebook works well in both Python 2 and 3:"
    ]
@@ -22,7 +28,9 @@
    "cell_type": "code",
    "execution_count": 1,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -31,7 +39,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "# Vectors\n",
     "## Definition\n",
@@ -73,7 +84,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Vectors in python\n",
     "In python, a vector can be represented in many ways, the simplest being a regular python list of numbers:"
@@ -83,7 +97,9 @@
    "cell_type": "code",
    "execution_count": 2,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -92,7 +108,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Since we plan to do quite a lot of scientific calculations, it is much better to use NumPy's `ndarray`, which provides a lot of convenient and optimized implementations of essential mathematical operations on vectors (for more details about NumPy, check out the [NumPy tutorial](tools_numpy.ipynb)). For example:"
    ]
@@ -101,7 +120,9 @@
    "cell_type": "code",
    "execution_count": 3,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -112,7 +133,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The size of a vector can be obtained using the `size` attribute:"
    ]
@@ -121,7 +145,9 @@
    "cell_type": "code",
    "execution_count": 4,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -130,7 +156,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The $i^{th}$ element (also called *entry* or *item*) of a vector $\\textbf{v}$ is noted $\\textbf{v}_i$.\n",
     "\n",
@@ -141,7 +170,9 @@
    "cell_type": "code",
    "execution_count": 5,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -150,7 +181,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Plotting vectors\n",
     "To plot vectors we will use matplotlib, so let's start by importing it (for details about matplotlib, check the [matplotlib tutorial](tools_matplotlib.ipynb)):"
@@ -160,7 +194,9 @@
    "cell_type": "code",
    "execution_count": 6,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -170,7 +206,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### 2D vectors\n",
     "Let's create a couple very simple 2D vectors to plot:"
@@ -180,7 +219,9 @@
    "cell_type": "code",
    "execution_count": 7,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -190,7 +231,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "These vectors each have 2 elements, so they can easily be represented graphically on a 2D graph, for example as points:"
    ]
@@ -199,7 +243,9 @@
    "cell_type": "code",
    "execution_count": 8,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -212,7 +258,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Vectors can also be represented as arrows. Let's create a small convenience function to draw nice arrows:"
    ]
@@ -221,7 +270,9 @@
    "cell_type": "code",
    "execution_count": 9,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -233,7 +284,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Now let's draw the vectors **u** and **v** as arrows:"
    ]
@@ -242,7 +296,9 @@
    "cell_type": "code",
    "execution_count": 10,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -255,7 +311,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### 3D vectors\n",
     "Plotting 3D vectors is also relatively straightforward. First let's create two 3D vectors:"
@@ -265,7 +324,9 @@
    "cell_type": "code",
    "execution_count": 11,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -275,7 +336,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Now let's plot them using matplotlib's `Axes3D`:"
    ]
@@ -284,7 +348,9 @@
    "cell_type": "code",
    "execution_count": 12,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -299,7 +365,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "It is a bit hard to visualize exactly where in space these two points are, so let's add vertical lines. We'll create a small convenience function to plot a list of 3d vectors with vertical lines attached:"
    ]
@@ -308,7 +377,9 @@
    "cell_type": "code",
    "execution_count": 13,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -327,7 +398,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Norm\n",
     "The norm of a vector $\\textbf{u}$, noted $\\left \\Vert \\textbf{u} \\right \\|$, is a measure of the length (a.k.a. the magnitude) of $\\textbf{u}$. There are multiple possible norms, but the most common one (and the only one we will discuss here) is the Euclidian norm, which is defined as:\n",
@@ -341,7 +415,9 @@
    "cell_type": "code",
    "execution_count": 14,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -355,7 +431,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "However, it is much more efficient to use NumPy's `norm` function, available in the `linalg` (**Lin**ear **Alg**ebra) module:"
    ]
@@ -364,7 +443,9 @@
    "cell_type": "code",
    "execution_count": 15,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -374,7 +455,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Let's plot a little diagram to confirm that the length of vector $\\textbf{v}$ is indeed $\\approx5.4$:"
    ]
@@ -383,7 +467,9 @@
    "cell_type": "code",
    "execution_count": 16,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -397,14 +483,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Looks about right!"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Addition\n",
     "Vectors of same size can be added together. Addition is performed *elementwise*:"
@@ -414,7 +506,9 @@
    "cell_type": "code",
    "execution_count": 17,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -426,7 +520,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Let's look at what vector addition looks like graphically:"
    ]
@@ -436,6 +533,8 @@
    "execution_count": 18,
    "metadata": {
     "collapsed": false,
+    "deletable": true,
+    "editable": true,
     "scrolled": true
    },
    "outputs": [],
@@ -457,7 +556,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Vector addition is **commutative**, meaning that $\\textbf{u} + \\textbf{v} = \\textbf{v} + \\textbf{u}$. You can see it on the previous image: following $\\textbf{u}$ *then* $\\textbf{v}$ leads to the same point as following $\\textbf{v}$ *then* $\\textbf{u}$.\n",
     "\n",
@@ -466,7 +568,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "If you have a shape defined by a number of points (vectors), and you add a vector $\\textbf{v}$ to all of these points, then the whole shape gets shifted by $\\textbf{v}$. This is called a [geometric translation](https://en.wikipedia.org/wiki/Translation_%28geometry%29):"
    ]
@@ -475,7 +580,9 @@
    "cell_type": "code",
    "execution_count": 19,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -508,14 +615,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Finally, substracting a vector is like adding the opposite vector."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Multiplication by a scalar\n",
     "Vectors can be multiplied by scalars. All elements in the vector are multiplied by that number, for example:"
@@ -525,7 +638,9 @@
    "cell_type": "code",
    "execution_count": 20,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -536,7 +651,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Graphically, scalar multiplication results in changing the scale of a figure, hence the name *scalar*. The distance from the origin (the point at coordinates equal to zero) is also multiplied by the scalar. For example, let's scale up by a factor of `k = 2.5`:"
    ]
@@ -545,7 +663,9 @@
    "cell_type": "code",
    "execution_count": 21,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -574,7 +694,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "As you might guess, dividing a vector by a scalar is equivalent to multiplying by its inverse:\n",
     "\n",
@@ -583,7 +706,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Scalar multiplication is **commutative**: $\\lambda \\times \\textbf{u} = \\textbf{u} \\times \\lambda$.\n",
     "\n",
@@ -594,7 +720,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Zero, unit and normalized vectors\n",
     "* A **zero-vector ** is a vector full of 0s.\n",
@@ -607,7 +736,9 @@
    "cell_type": "code",
    "execution_count": 22,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -624,7 +755,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Dot product\n",
     "### Definition\n",
@@ -646,7 +780,9 @@
    "cell_type": "code",
    "execution_count": 23,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -658,7 +794,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "But a *much* more efficient implementation is provided by NumPy with the `dot` function:"
    ]
@@ -667,7 +806,9 @@
    "cell_type": "code",
    "execution_count": 24,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -676,7 +817,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Equivalently, you can use the `dot` method of `ndarray`s:"
    ]
@@ -685,7 +829,9 @@
    "cell_type": "code",
    "execution_count": 25,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -694,7 +840,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "**Caution**: the `*` operator will perform an *elementwise* multiplication, *NOT* a dot product:"
    ]
@@ -703,7 +852,9 @@
    "cell_type": "code",
    "execution_count": 26,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -716,7 +867,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Main properties\n",
     "* The dot product is **commutative**: $\\textbf{u} \\cdot \\textbf{v} = \\textbf{v} \\cdot \\textbf{u}$.\n",
@@ -728,7 +882,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Calculating the angle between vectors\n",
     "One of the many uses of the dot product is to calculate the angle between two non-zero vectors. Looking at the dot product definition, we can deduce the following formula:\n",
@@ -744,7 +901,9 @@
    "cell_type": "code",
    "execution_count": 27,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -759,14 +918,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Note: due to small floating point errors, `cos_theta` may be very slightly outside of the $[-1, 1]$ interval, which would make `arccos` fail. This is why we clipped the value within the range, using NumPy's `clip` function."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Projecting a point onto an axis\n",
     "The dot product is also very useful to project points onto an axis. The projection of vector $\\textbf{v}$ onto $\\textbf{u}$'s axis is given by this formula:\n",
@@ -782,7 +947,9 @@
    "cell_type": "code",
    "execution_count": 28,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -808,7 +975,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "# Matrices\n",
     "A matrix is a rectangular array of scalars (ie. any number: integer, real or complex) arranged in rows and columns, for example:\n",
@@ -822,7 +992,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Matrices in python\n",
     "In python, a matrix can be represented in various ways. The simplest is just a list of python lists:"
@@ -832,7 +1005,9 @@
    "cell_type": "code",
    "execution_count": 29,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -844,7 +1019,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "A much more efficient way is to use the NumPy library which provides optimized implementations of many matrix operations:"
    ]
@@ -853,7 +1031,9 @@
    "cell_type": "code",
    "execution_count": 30,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -866,7 +1046,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "By convention matrices generally have uppercase names, such as $A$.\n",
     "\n",
@@ -875,7 +1058,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Size\n",
     "The size of a matrix is defined by its number of rows and number of columns. It is noted $rows \\times columns$. For example, the matrix $A$ above is an example of a $2 \\times 3$ matrix: 2 rows, 3 columns. Caution: a $3 \\times 2$ matrix would have 3 rows and 2 columns.\n",
@@ -887,7 +1073,9 @@
    "cell_type": "code",
    "execution_count": 31,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -896,7 +1084,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "**Caution**: the `size` attribute represents the number of elements in the `ndarray`, not the matrix's size:"
    ]
@@ -905,7 +1096,9 @@
    "cell_type": "code",
    "execution_count": 32,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -914,7 +1107,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Element indexing\n",
     "The number located in the $i^{th}$ row, and $j^{th}$ column of a matrix $X$ is sometimes noted $X_{i,j}$ or $X_{ij}$, but there is no standard notation, so people often prefer to explicitely name the elements, like this: \"*let $X = (x_{i,j})_{1 ≤ i ≤ m, 1 ≤ j ≤ n}$*\". This means that $X$ is equal to:\n",
@@ -934,7 +1130,9 @@
    "cell_type": "code",
    "execution_count": 33,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -943,7 +1141,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The $i^{th}$ row vector is sometimes noted $M_i$ or $M_{i,*}$, but again there is no standard notation so people often prefer to explicitely define their own names, for example: \"*let **x**$_{i}$ be the $i^{th}$ row vector of matrix $X$*\". We will use the $M_{i,*}$, for the same reason as above. For example, to access $A_{2,*}$ (ie. $A$'s 2nd row vector):"
    ]
@@ -952,7 +1153,9 @@
    "cell_type": "code",
    "execution_count": 34,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -961,7 +1164,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Similarly, the $j^{th}$ column vector is sometimes noted $M^j$ or $M_{*,j}$, but there is no standard notation. We will use $M_{*,j}$. For example, to access $A_{*,3}$ (ie. $A$'s 3rd column vector):"
    ]
@@ -970,7 +1176,9 @@
    "cell_type": "code",
    "execution_count": 35,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -979,7 +1187,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Note that the result is actually a one-dimensional NumPy array: there is no such thing as a *vertical* or *horizontal* one-dimensional array. If you need to actually represent a row vector as a one-row matrix (ie. a 2D NumPy array), or a column vector as a one-column matrix, then you need to use a slice instead of an integer when accessing the row or column, for example:"
    ]
@@ -988,7 +1199,9 @@
    "cell_type": "code",
    "execution_count": 36,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -999,7 +1212,9 @@
    "cell_type": "code",
    "execution_count": 37,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1008,7 +1223,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Square, triangular, diagonal and identity matrices\n",
     "A **square matrix** is a matrix that has the same number of rows and columns, for example a $3 \\times 3$ matrix:\n",
@@ -1022,7 +1240,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "An **upper triangular matrix** is a special kind of square matrix where all the elements *below* the main diagonal (top-left to bottom-right) are zero, for example:\n",
     "\n",
@@ -1035,7 +1256,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Similarly, a **lower triangular matrix** is a square matrix where all elements *above* the main diagonal are zero, for example:\n",
     "\n",
@@ -1048,14 +1272,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "A **triangular matrix** is one that is either lower triangular or upper triangular."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "A matrix that is both upper and lower triangular is called a **diagonal matrix**, for example:\n",
     "\n",
@@ -1072,7 +1302,9 @@
    "cell_type": "code",
    "execution_count": 38,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1081,7 +1313,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "If you pass a matrix to the `diag` function, it will happily extract the diagonal values:"
    ]
@@ -1090,7 +1325,9 @@
    "cell_type": "code",
    "execution_count": 39,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1104,7 +1341,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Finally, the **identity matrix** of size $n$, noted $I_n$, is a diagonal matrix of size $n \\times n$ with $1$'s in the main diagonal, for example $I_3$:\n",
     "\n",
@@ -1121,7 +1361,9 @@
    "cell_type": "code",
    "execution_count": 40,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1130,14 +1372,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The identity matrix is often noted simply $I$ (instead of $I_n$) when its size is clear given the context. It is called the *identity* matrix because multiplying a matrix with it leaves the matrix unchanged as we will see below."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Adding matrices\n",
     "If two matrices $Q$ and $R$ have the same size $m \\times n$, they can be added together. Addition is performed *elementwise*: the result is also a $m \\times n$ matrix $S$ where each element is the sum of the elements at the corresponding position: $S_{i,j} = Q_{i,j} + R_{i,j}$\n",
@@ -1158,7 +1406,9 @@
    "cell_type": "code",
    "execution_count": 41,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1170,7 +1420,9 @@
    "cell_type": "code",
    "execution_count": 42,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1181,7 +1433,9 @@
    "cell_type": "code",
    "execution_count": 43,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1190,7 +1444,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "**Addition is *commutative***, meaning that $A + B = B + A$:"
    ]
@@ -1199,7 +1456,9 @@
    "cell_type": "code",
    "execution_count": 44,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1208,7 +1467,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "**It is also *associative***, meaning that $A + (B + C) = (A + B) + C$:"
    ]
@@ -1217,7 +1479,9 @@
    "cell_type": "code",
    "execution_count": 45,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1230,7 +1494,9 @@
    "cell_type": "code",
    "execution_count": 46,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1239,7 +1505,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Scalar multiplication\n",
     "A matrix $M$ can be multiplied by a scalar $\\lambda$. The result is noted $\\lambda M$, and it is a matrix of the same size as $M$ with all elements multiplied by $\\lambda$:\n",
@@ -1264,7 +1533,9 @@
    "cell_type": "code",
    "execution_count": 47,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1273,7 +1544,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Scalar multiplication is also defined on the right hand side, and gives the same result: $M \\lambda = \\lambda M$. For example:"
    ]
@@ -1282,7 +1556,9 @@
    "cell_type": "code",
    "execution_count": 48,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1291,7 +1567,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "This makes scalar multiplication **commutative**.\n",
     "\n",
@@ -1302,7 +1581,9 @@
    "cell_type": "code",
    "execution_count": 49,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1313,7 +1594,9 @@
    "cell_type": "code",
    "execution_count": 50,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1322,7 +1605,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Finally, it is **distributive over addition** of matrices, meaning that $\\lambda (Q + R) = \\lambda Q + \\lambda R$:"
    ]
@@ -1331,7 +1617,9 @@
    "cell_type": "code",
    "execution_count": 51,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1342,7 +1630,9 @@
    "cell_type": "code",
    "execution_count": 52,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1351,7 +1641,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Matrix multiplication\n",
     "So far, matrix operations have been rather intuitive. But multiplying matrices is a bit more involved.\n",
@@ -1396,7 +1689,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Let's multiply two matrices in NumPy, using `ndarray`'s `dot` method:\n",
     "\n",
@@ -1419,7 +1715,9 @@
    "cell_type": "code",
    "execution_count": 53,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1434,7 +1732,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Let's check this result by looking at one element, just to be sure: looking at $E_{2,3}$ for example, we need to multiply elements in $A$'s $2^{nd}$ row by elements in $D$'s $3^{rd}$ column, and sum up these products:"
    ]
@@ -1443,7 +1744,9 @@
    "cell_type": "code",
    "execution_count": 54,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1454,7 +1757,9 @@
    "cell_type": "code",
    "execution_count": 55,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1463,7 +1768,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Looks good! You can check the other elements until you get used to the algorithm.\n",
     "\n",
@@ -1474,7 +1782,9 @@
    "cell_type": "code",
    "execution_count": 56,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1486,7 +1796,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "This illustrates the fact that **matrix multiplication is *NOT* commutative**: in general $QR ≠ RQ$\n",
     "\n",
@@ -1497,7 +1810,9 @@
    "cell_type": "code",
    "execution_count": 57,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1513,7 +1828,9 @@
    "cell_type": "code",
    "execution_count": 58,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1522,7 +1839,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "On the other hand, **matrix multiplication *is* associative**, meaning that $Q(RS) = (QR)S$. Let's create a $4 \\times 5$ matrix $G$ to illustrate this:"
    ]
@@ -1531,7 +1851,9 @@
    "cell_type": "code",
    "execution_count": 59,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1547,7 +1869,9 @@
    "cell_type": "code",
    "execution_count": 60,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1556,7 +1880,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "It is also ***distributive* over addition** of matrices, meaning that $(Q + R)S = QS + RS$. For example:"
    ]
@@ -1565,7 +1892,9 @@
    "cell_type": "code",
    "execution_count": 61,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1576,7 +1905,9 @@
    "cell_type": "code",
    "execution_count": 62,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1585,7 +1916,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The product of a matrix $M$ by the identity matrix (of matching size) results in the same matrix $M$. More formally, if $M$ is an $m \\times n$ matrix, then:\n",
     "\n",
@@ -1602,7 +1936,9 @@
    "cell_type": "code",
    "execution_count": 63,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1613,7 +1949,9 @@
    "cell_type": "code",
    "execution_count": 64,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1622,7 +1960,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "**Caution**: NumPy's `*` operator performs elementwise multiplication, *NOT* a matrix multiplication:"
    ]
@@ -1632,6 +1973,8 @@
    "execution_count": 65,
    "metadata": {
     "collapsed": false,
+    "deletable": true,
+    "editable": true,
     "scrolled": true
    },
    "outputs": [],
@@ -1641,7 +1984,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "**The @ infix operator**\n",
     "\n",
@@ -1652,7 +1998,9 @@
    "cell_type": "code",
    "execution_count": 66,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1668,14 +2016,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Note: `Q @ R` is actually equivalent to `Q.__matmul__(R)` which is implemented by NumPy as `np.matmul(Q, R)`, not as `Q.dot(R)`. The main difference is that `matmul` does not support scalar multiplication, while `dot` does, so you can write `Q.dot(3)`, which is equivalent to `Q * 3`, but you cannot write `Q @ 3` ([more details](http://stackoverflow.com/a/34142617/38626))."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Matrix transpose\n",
     "The transpose of a matrix $M$ is a matrix noted $M^T$ such that the $i^{th}$ row in $M^T$ is equal to the $i^{th}$ column in $M$:\n",
@@ -1704,7 +2058,9 @@
    "cell_type": "code",
    "execution_count": 67,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1715,7 +2071,9 @@
    "cell_type": "code",
    "execution_count": 68,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1724,7 +2082,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "As you might expect, transposing a matrix twice returns the original matrix:"
    ]
@@ -1733,7 +2094,9 @@
    "cell_type": "code",
    "execution_count": 69,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1742,7 +2105,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Transposition is distributive over addition of matrices, meaning that $(Q + R)^T = Q^T + R^T$. For example:"
    ]
@@ -1751,7 +2117,9 @@
    "cell_type": "code",
    "execution_count": 70,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1762,7 +2130,9 @@
    "cell_type": "code",
    "execution_count": 71,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1771,7 +2141,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Moreover, $(Q \\cdot R)^T = R^T \\cdot Q^T$. Note that the order is reversed. For example:"
    ]
@@ -1780,7 +2153,9 @@
    "cell_type": "code",
    "execution_count": 72,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1791,7 +2166,9 @@
    "cell_type": "code",
    "execution_count": 73,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1800,7 +2177,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "A **symmetric matrix** $M$ is defined as a matrix that is equal to its transpose: $M^T = M$. This definition implies that it must be a square matrix whose elements are symmetric relative to the main diagonal, for example:\n",
     "\n",
@@ -1818,7 +2198,9 @@
    "cell_type": "code",
    "execution_count": 74,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1828,7 +2210,9 @@
   {
    "cell_type": "markdown",
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "deletable": true,
+    "editable": true
    },
    "source": [
     "## Converting 1D arrays to 2D arrays in NumPy\n",
@@ -1839,7 +2223,9 @@
    "cell_type": "code",
    "execution_count": 75,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1850,7 +2236,9 @@
    "cell_type": "code",
    "execution_count": 76,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1859,7 +2247,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "We want to convert $\\textbf{u}$ into a row vector before transposing it. There are a few ways to do this:"
    ]
@@ -1868,7 +2259,9 @@
    "cell_type": "code",
    "execution_count": 77,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1878,7 +2271,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Notice the extra square brackets: this is a 2D array with just one row (ie. a 1x2 matrix). In other words it really is a **row vector**."
    ]
@@ -1887,7 +2283,9 @@
    "cell_type": "code",
    "execution_count": 78,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1896,7 +2294,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "This quite explicit: we are asking for a new vertical axis, keeping the existing data as the horizontal axis."
    ]
@@ -1905,7 +2306,9 @@
    "cell_type": "code",
    "execution_count": 79,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1914,7 +2317,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "This is equivalent, but a little less explicit."
    ]
@@ -1923,7 +2329,9 @@
    "cell_type": "code",
    "execution_count": 80,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1932,7 +2340,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "This is the shortest version, but you probably want to avoid it because it is unclear. The reason it works is that `np.newaxis` is actually equal to `None`, so this is equivalent to the previous version.\n",
     "\n",
@@ -1943,7 +2354,9 @@
    "cell_type": "code",
    "execution_count": 81,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1952,7 +2365,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Great! We now have a nice **column vector**.\n",
     "\n",
@@ -1963,7 +2379,9 @@
    "cell_type": "code",
    "execution_count": 82,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1972,7 +2390,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Plotting a matrix\n",
     "We have already seen that vectors can been represented as points or arrows in N-dimensional space. Is there a good graphical representation of matrices? Well you can simply see a matrix as a list of vectors, so plotting a matrix results in many points or arrows. For example, let's create a $2 \\times 4$ matrix `P` and plot it as points:"
@@ -1982,7 +2403,9 @@
    "cell_type": "code",
    "execution_count": 83,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -1998,7 +2421,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Of course we could also have stored the same 4 vectors as row vectors instead of column vectors, resulting in a $4 \\times 2$ matrix (the transpose of $P$, in fact). It is really an arbitrary choice.\n",
     "\n",
@@ -2009,7 +2435,9 @@
    "cell_type": "code",
    "execution_count": 84,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2022,7 +2450,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Or you can represent it as a polygon: matplotlib's `Polygon` class expects an $n \\times 2$ NumPy array, not a $2 \\times n$ array, so we just need to give it $P^T$:"
    ]
@@ -2031,7 +2462,9 @@
    "cell_type": "code",
    "execution_count": 85,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2044,7 +2477,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Geometric applications of matrix operations\n",
     "We saw earlier that vector addition results in a geometric translation, vector multiplication by a scalar results in rescaling (zooming in or out, centered on the origin), and vector dot product results in projecting a vector onto another vector, rescaling and measuring the resulting coordinate.\n",
@@ -2054,7 +2490,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Addition = multiple geometric translations\n",
     "First, adding two matrices together is equivalent to adding all their vectors together. For example, let's create a $2 \\times 4$ matrix $H$ and add it to $P$, and look at the result:"
@@ -2064,7 +2503,9 @@
    "cell_type": "code",
    "execution_count": 86,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2093,7 +2534,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "If we add a matrix full of identical vectors, we get a simple geometric translation:"
    ]
@@ -2102,7 +2546,9 @@
    "cell_type": "code",
    "execution_count": 87,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2124,7 +2570,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Although matrices can only be added together if they have the same size, NumPy allows adding a row vector or a column vector to a matrix: this is called *broadcasting* and is explained in further details in the [NumPy tutorial](tools_numpy.ipynb). We could have obtained the same result as above with:"
    ]
@@ -2133,7 +2582,9 @@
    "cell_type": "code",
    "execution_count": 88,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2142,7 +2593,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Scalar multiplication\n",
     "Multiplying a matrix by a scalar results in all its vectors being multiplied by that scalar, so unsurprisingly, the geometric result is a rescaling of the entire figure. For example, let's rescale our polygon by a factor of 60% (zooming out, centered on the origin):"
@@ -2152,7 +2606,9 @@
    "cell_type": "code",
    "execution_count": 89,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2175,7 +2631,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Matrix multiplication – Projection onto an axis\n",
     "Matrix multiplication is more complex to visualize, but it is also the most powerful tool in the box.\n",
@@ -2187,7 +2646,9 @@
    "cell_type": "code",
    "execution_count": 90,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2196,7 +2657,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Now let's look at the dot product $P \\cdot U$:"
    ]
@@ -2205,7 +2669,9 @@
    "cell_type": "code",
    "execution_count": 91,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2214,7 +2680,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "These are the horizontal coordinates of the vectors in $P$. In other words, we just projected $P$ onto the horizontal axis:"
    ]
@@ -2223,7 +2692,9 @@
    "cell_type": "code",
    "execution_count": 92,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2248,7 +2719,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "We can actually project on any other axis by just replacing $U$ with any other unit vector. For example, let's project on the axis that is at a 30° angle above the horizontal axis:"
    ]
@@ -2257,7 +2731,9 @@
    "cell_type": "code",
    "execution_count": 93,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2269,14 +2745,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Good! Remember that the dot product of a unit vector and a matrix basically performs a projection on an axis and gives us the coordinates of the resulting points on that axis."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Matrix multiplication – Rotation\n",
     "Now let's create a $2 \\times 2$ matrix $V$ containing two unit vectors that make 30° and 120° angles with the horizontal axis:\n",
@@ -2288,7 +2770,9 @@
    "cell_type": "code",
    "execution_count": 94,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2302,7 +2786,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Let's look at the product $VP$:"
    ]
@@ -2311,7 +2798,9 @@
    "cell_type": "code",
    "execution_count": 95,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2320,7 +2809,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The first row is equal to $V_{1,*} P$, which is the coordinates of the projection of $P$ onto the 30° axis, as we have seen above. The second row is $V_{2,*} P$, which is the coordinates of the projection of $P$ onto the 120° axis. So basically we obtained the coordinates of $P$ after rotating the horizontal and vertical axes by 30° (or equivalently after rotating the polygon by -30° around the origin)! Let's plot $VP$ to see this:"
    ]
@@ -2329,7 +2821,9 @@
    "cell_type": "code",
    "execution_count": 96,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2340,14 +2834,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Matrix $V$ is called a **rotation matrix**."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "### Matrix multiplication – Other linear transformations\n",
     "More generally, any linear transformation $f$ that maps n-dimensional vectors to m-dimensional vectors can be represented as an $m \\times n$ matrix. For example, say $\\textbf{u}$ is a 3-dimensional vector:\n",
@@ -2384,6 +2884,8 @@
    "execution_count": 97,
    "metadata": {
     "collapsed": false,
+    "deletable": true,
+    "editable": true,
     "scrolled": true
    },
    "outputs": [],
@@ -2399,7 +2901,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Let's look at how this transformation affects the **unit square**: "
    ]
@@ -2408,7 +2913,9 @@
    "cell_type": "code",
    "execution_count": 98,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2423,7 +2930,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Now let's look at a **squeeze mapping**:"
    ]
@@ -2432,7 +2942,9 @@
    "cell_type": "code",
    "execution_count": 99,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2447,7 +2959,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The effect on the unit square is:"
    ]
@@ -2456,7 +2971,9 @@
    "cell_type": "code",
    "execution_count": 100,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2467,7 +2984,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Let's show a last one: reflection through the horizontal axis:"
    ]
@@ -2476,7 +2996,9 @@
    "cell_type": "code",
    "execution_count": 101,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2491,7 +3013,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Matrix inverse\n",
     "Now that we understand that a matrix can represent any linear transformation, a natural question is: can we find a transformation matrix that reverses the effect of a given transformation matrix $F$? The answer is yes… sometimes! When it exists, such a matrix is called the **inverse** of $F$, and it is noted $F^{-1}$.\n",
@@ -2503,7 +3028,9 @@
    "cell_type": "code",
    "execution_count": 102,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2521,7 +3048,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "We applied a shear mapping on $P$, just like we did before, but then we applied a second transformation to the result, and *lo and behold* this had the effect of coming back to the original $P$ (we plotted the original $P$'s outline to double check). The second transformation is the inverse of the first one.\n",
     "\n",
@@ -2532,7 +3062,9 @@
    "cell_type": "code",
    "execution_count": 103,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2542,7 +3074,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Only square matrices can be inversed. This makes sense when you think about it: if you have a transformation that reduces the number of dimensions, then some information is lost and there is no way that you can get it back. For example say you use a $2 \\times 3$ matrix to project a 3D object onto a plane. The result may look like this:"
    ]
@@ -2551,7 +3086,9 @@
    "cell_type": "code",
    "execution_count": 104,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2564,7 +3101,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Looking at this image, it is impossible to tell whether this is the projection of a cube or the projection of a narrow rectangular object. Some information has been lost in the projection.\n",
     "\n",
@@ -2575,7 +3115,9 @@
    "cell_type": "code",
    "execution_count": 105,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2590,7 +3132,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "This transformation matrix performs a projection onto the horizontal axis. Our polygon gets entirely flattened out so some information is entirely lost and it is impossible to go back to the original polygon using a linear transformation. In other words, $F_{project}$ has no inverse. Such a square matrix that cannot be inversed is called a **singular matrix** (aka degenerate matrix). If we ask NumPy to calculate its inverse, it raises an exception:"
    ]
@@ -2599,7 +3144,9 @@
    "cell_type": "code",
    "execution_count": 106,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2611,7 +3158,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Here is another example of a singular matrix. This one performs a projection onto the axis at a 30° angle above the horizontal axis:"
    ]
@@ -2620,7 +3170,9 @@
    "cell_type": "code",
    "execution_count": 107,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2636,7 +3188,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "But this time, due to floating point rounding errors, NumPy manages to calculate an inverse (notice how large the elements are, though):"
    ]
@@ -2645,7 +3200,9 @@
    "cell_type": "code",
    "execution_count": 108,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2654,7 +3211,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "As you might expect, the dot product of a matrix by its inverse results in the identity matrix:\n",
     "\n",
@@ -2667,7 +3227,9 @@
    "cell_type": "code",
    "execution_count": 109,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2676,7 +3238,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Another way to express this is that the inverse of the inverse of a matrix $M$ is $M$ itself:\n",
     "\n",
@@ -2687,7 +3252,9 @@
    "cell_type": "code",
    "execution_count": 110,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2696,7 +3263,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Also, the inverse of scaling by a factor of $\\lambda$ is of course scaling by a factor or $\\frac{1}{\\lambda}$:\n",
     "\n",
@@ -2711,7 +3281,9 @@
    "cell_type": "code",
    "execution_count": 111,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2726,7 +3298,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Finally, a square matrix $H$ whose inverse is its own transpose is an **orthogonal matrix**:\n",
     "\n",
@@ -2743,7 +3318,9 @@
    "cell_type": "code",
    "execution_count": 112,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2752,7 +3329,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Determinant\n",
     "The determinant of a square matrix $M$, noted $\\det(M)$ or $\\det M$ or $|M|$ is a value that can be calculated from its elements $(M_{i,j})$ using various equivalent methods. One of the simplest methods is this recursive approach:\n",
@@ -2790,7 +3370,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "To get the determinant of a matrix, you can call NumPy's `det` function in the `numpy.linalg` module:"
    ]
@@ -2799,7 +3382,9 @@
    "cell_type": "code",
    "execution_count": 113,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2813,7 +3398,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "One of the main uses of the determinant is to *determine* whether a square matrix can be inversed or not: if the determinant is equal to 0, then the matrix *cannot* be inversed (it is a singular matrix), and if the determinant is not 0, then it *can* be inversed.\n",
     "\n",
@@ -2824,7 +3412,9 @@
    "cell_type": "code",
    "execution_count": 114,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2833,7 +3423,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "That's right, $F_{project}$ is singular, as we saw earlier."
    ]
@@ -2842,7 +3435,9 @@
    "cell_type": "code",
    "execution_count": 115,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2851,7 +3446,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "This determinant is suspiciously close to 0: it really should be 0, but it's not due to tiny floating point errors. The matrix is actually singular."
    ]
@@ -2860,7 +3458,9 @@
    "cell_type": "code",
    "execution_count": 116,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2869,14 +3469,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Perfect! This matrix *can* be inversed as we saw earlier. Wow, math really works!"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The determinant can also be used to measure how much a linear transformation affects surface areas: for example, the projection matrices $F_{project}$ and $F_{project\\_30}$ completely flatten the polygon $P$, until its area is zero. This is why the determinant of these matrices is 0. The shear mapping modified the shape of the polygon, but it did not affect its surface area, which is why the determinant is 1. You can try computing the determinant of a rotation matrix, and you should also find 1. What about a scaling matrix? Let's see:"
    ]
@@ -2885,7 +3491,9 @@
    "cell_type": "code",
    "execution_count": 117,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2900,7 +3508,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "We rescaled the polygon by a factor of 1/2 on both vertical and horizontal axes so the surface area of the resulting polygon is 1/4$^{th}$ of the original polygon. Let's compute the determinant and check that:"
    ]
@@ -2909,7 +3520,9 @@
    "cell_type": "code",
    "execution_count": 118,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2918,7 +3531,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Correct!\n",
     "\n",
@@ -2929,7 +3545,9 @@
    "cell_type": "code",
    "execution_count": 119,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2938,7 +3556,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Composing linear transformations\n",
     "Several linear transformations can be chained simply by performing multiple dot products in a row. For example, to perform a squeeze mapping followed by a shear mapping, just write:"
@@ -2948,7 +3569,9 @@
    "cell_type": "code",
    "execution_count": 120,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2957,7 +3580,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Since the dot product is associative, the following code is equivalent:"
    ]
@@ -2966,7 +3592,9 @@
    "cell_type": "code",
    "execution_count": 121,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2975,7 +3603,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Note that the order of the transformations is the reverse of the dot product order.\n",
     "\n",
@@ -2986,7 +3617,9 @@
    "cell_type": "code",
    "execution_count": 122,
    "metadata": {
-    "collapsed": true
+    "collapsed": true,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -2996,14 +3629,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "From now on we can perform both transformations in just one dot product, which can lead to a very significant performance boost."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "What if you want to perform the inverse of this double transformation? Well, if you squeezed and then you sheared, and you want to undo what you have done, it should be obvious that you should unshear first and then unsqueeze. In more mathematical terms, given two invertible (aka nonsingular) matrices $Q$ and $R$:\n",
     "\n",
@@ -3016,7 +3655,9 @@
    "cell_type": "code",
    "execution_count": 123,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3025,7 +3666,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Singular Value Decomposition\n",
     "It turns out that any $m \\times n$ matrix $M$ can be decomposed into the dot product of three simple matrices:\n",
@@ -3042,7 +3686,9 @@
    "cell_type": "code",
    "execution_count": 124,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3054,7 +3700,9 @@
    "cell_type": "code",
    "execution_count": 125,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3063,7 +3711,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Note that this is just a 1D array containing the diagonal values of Σ. To get the actual matrix Σ, we can use NumPy's `diag` function:"
    ]
@@ -3072,7 +3723,9 @@
    "cell_type": "code",
    "execution_count": 126,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3082,7 +3735,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Now let's check that $U \\cdot \\Sigma \\cdot V^T$ is indeed equal to `F_shear`:"
    ]
@@ -3091,7 +3747,9 @@
    "cell_type": "code",
    "execution_count": 127,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3102,7 +3760,9 @@
    "cell_type": "code",
    "execution_count": 128,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3111,7 +3771,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "It worked like a charm. Let's apply these transformations one by one (in reverse order) on the unit square to understand what's going on. First, let's apply the first rotation $V^T$:"
    ]
@@ -3120,7 +3783,9 @@
    "cell_type": "code",
    "execution_count": 129,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3131,7 +3796,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Now let's rescale along the vertical and horizontal axes using $\\Sigma$:"
    ]
@@ -3140,7 +3808,9 @@
    "cell_type": "code",
    "execution_count": 130,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3151,7 +3821,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Finally, we apply the second rotation $U$:"
    ]
@@ -3160,7 +3833,9 @@
    "cell_type": "code",
    "execution_count": 131,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3171,14 +3846,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "And we can see that the result is indeed a shear mapping of the original unit square."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Eigenvectors and eigenvalues\n",
     "An **eigenvector** of a square matrix $M$ (also called a **characteristic vector**) is a non-zero vector that remains on the same line after transformation by the linear transformation associated with $M$. A more formal definition is any vector $v$ such that:\n",
@@ -3200,7 +3881,9 @@
    "cell_type": "code",
    "execution_count": 132,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3212,7 +3895,9 @@
    "cell_type": "code",
    "execution_count": 133,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3221,7 +3906,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Indeed the horizontal vectors are stretched by a factor of 1.4, and the vertical vectors are shrunk by a factor of 1/1.4=0.714…, so far so good. Let's look at the shear mapping matrix $F_{shear}$:"
    ]
@@ -3230,7 +3918,9 @@
    "cell_type": "code",
    "execution_count": 134,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3242,7 +3932,9 @@
    "cell_type": "code",
    "execution_count": 135,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3251,14 +3943,20 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "Wait, what!? We expected just one unit eigenvector, not two. The second vector is almost equal to $\\begin{pmatrix}-1 \\\\ 0 \\end{pmatrix}$, which is on the same line as the first vector $\\begin{pmatrix}1 \\\\ 0 \\end{pmatrix}$. This is due to floating point errors. We can safely ignore vectors that are (almost) colinear (ie. on the same line)."
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "## Trace\n",
     "The trace of a square matrix $M$, noted $tr(M)$ is the sum of the values on its main diagonal. For example:"
@@ -3268,7 +3966,9 @@
    "cell_type": "code",
    "execution_count": 136,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3282,7 +3982,10 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "The trace does not have a simple geometric interpretation (in general), but it has a number of properties that make it useful in many areas:\n",
     "* $tr(A + B) = tr(A) + tr(B)$\n",
@@ -3298,7 +4001,9 @@
    "cell_type": "code",
    "execution_count": 137,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "deletable": true,
+    "editable": true
    },
    "outputs": [],
    "source": [
@@ -3307,30 +4012,42 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
    "source": [
     "# What next?\n",
-    "This concludes this introduction to Linear Algeabra. Although these basics cover most of what you will need to know for Machine Learning, if you wish to go deeper into this topic there are many options available: Linear Algebra [books](http://linear.axler.net/), [Khan Academy](https://www.khanacademy.org/math/linear-algebra) lessons, or just [Wikipedia](https://en.wikipedia.org/wiki/Linear_algebra) pages. "
+    "This concludes this introduction to Linear Algebra. Although these basics cover most of what you will need to know for Machine Learning, if you wish to go deeper into this topic there are many options available: Linear Algebra [books](http://linear.axler.net/), [Khan Academy](https://www.khanacademy.org/math/linear-algebra) lessons, or just [Wikipedia](https://en.wikipedia.org/wiki/Linear_algebra) pages. "
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 2",
+   "display_name": "Python 3",
    "language": "python",
-   "name": "python2"
+   "name": "python3"
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   "language_info": {
    "codemirror_mode": {
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+    "version": 3
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+   "version": "3.5.3"
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    "toc_cell": false,