Exercise 10 - Vectors and Matrices.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Linear Algebra "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Representing vectors in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\vec{a}=\\begin{bmatrix} 3\\\\4\\end{bmatrix}; \\quad \n",
    "\\vec{b}=\\begin{bmatrix} 2\\\\-1\\end{bmatrix}; \\quad \n",
    "\\vec{c}=\\begin{bmatrix} -3\\\\-3\\end{bmatrix}; \\quad \n",
    "\\vec{d}=\\begin{bmatrix} -2\\\\3\\end{bmatrix}; \\quad\n",
    "\\vec{e}=\\begin{bmatrix} 2\\\\3\\\\5\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import `numpy` library"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By using the numpy *arrays*, represent the vectors $\\vec{a}$ to $\\vec{e}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Print the values of some of the vectors to make sure they match the definition:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use NumPy's built-in function `linalg.norm` to compute the length of the vectors $\\vec{a}$ and $\\vec{e}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector Operations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Adding, subtracting and multiplying vectors by using code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Add the vectors $\\vec{a}$ and $\\vec{b}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Print the result of the addition:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Subtract vector $\\vec{b}$ from $\\vec{a}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply the vector $\\vec{a}$ by some number (scalar):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Multiplying vectors elementwise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply vectors $\\vec{a}$ and $\\vec{b}$ elementwise:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Computing the dot product by using code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the dot product of $\\vec{a}$ and $\\vec{b}$ by multipling two vectors elementwise and summing the multiples. For summing, use the NumPy function `np.sum`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Apply the last operaton to compute the length of the vector $\\vec{a}$ instead of using the function `linalg.norm`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is part of a broader convention of orderly arranging numbers in a rectangular, table-like structure called a __matrix__. A variable containing a vector is represented with a small letter and an arrow above it. A variable containing a matrix is conventionally represented with a capital letter. For example, M is a matrix containing 12 numbers arranged in a particular way."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$M_{3\\times4}=\\underset{\\;\\\\\\mathbf{3 \\, rows \\, \\times \\, 4 \\, \\text{columns}}}{\\begin{bmatrix} 1 & 3 & -2 & 4\\\\2 & 3 & -1 & -1\\\\ -1 & 1 & 3 & 4\\end{bmatrix}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Elementary Matrix Operations in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us define matrices $A$, $B$ and $C$ as follows:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "A=\\begin{bmatrix}\n",
    "4 & 2 & -1 & 5 \\\\\n",
    "2 & 1 & 3 & -3 \\\\\n",
    "-2 & -3 & 1 & 4 \\\\\n",
    "\\end{bmatrix}; \\quad\n",
    "B=\\begin{bmatrix}\n",
    "-1 & 1 \\\\\n",
    "3 & -3 \\\\\n",
    "4 & -2 \\\\\n",
    "5 & 5 \\\\\n",
    "\\end{bmatrix}; \\quad\n",
    "C=\\begin{bmatrix}\n",
    "1 & 3  \\\\\n",
    "3 & -1 \\\\\n",
    "\\end{bmatrix}; \\quad\n",
    "D=\\begin{bmatrix}\n",
    "2 & 1  \\\\\n",
    "-1 & 2 \\\\\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use NumPy's data structure called array to represent matrix A:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we defined the other matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "B = np.array([[-1,  1],\n",
    "              [ 3, -3],\n",
    "              [ 4, -2],\n",
    "              [ 5,  5]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = np.array([[-1,  3],\n",
    "              [ 3, -1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "D = np.array([[ 2, 1],\n",
    "              [-1, 2]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Print the variables containing the matrices to make sure they match the definition:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Matrix Addition and Subtraction in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Add matrices $A$ and $B$ and analyse the result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Add the matrices $C$ and $D$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Add the matrix $A$ to itself:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Multiplying a Matrix by a Scalar in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply the matrix $A$ by 5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multiply the matrix $A$ by -5:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Matrix Multiplication in Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the matrix product $A\\times B$, by using the NumPy command `np.dot`, and analyse the result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the matrix product $B\\times A$, and analyse the result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the matrix products $B\\times C$ and $B\\times D$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing matrix products $C\\times D$ and $D\\times C$, and analyse the result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}