diff --git a/Notebooks/Chap01/1_1_BackgroundMathematics.ipynb b/Notebooks/Chap01/1_1_BackgroundMathematics.ipynb index 57cf726..f875714 100644 --- a/Notebooks/Chap01/1_1_BackgroundMathematics.ipynb +++ b/Notebooks/Chap01/1_1_BackgroundMathematics.ipynb @@ -1,16 +1,18 @@ { "cells": [ { + "attachments": {}, "cell_type": "markdown", "metadata": { - "id": "view-in-github", - "colab_type": "text" + "colab_type": "text", + "id": "view-in-github" }, "source": [ "\"Open" ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "s5zzKSOusPOB" @@ -39,6 +41,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "WV2Dl6owme2d" @@ -46,11 +49,11 @@ "source": [ "**Linear functions**
We will be using the term *linear equation* to mean a weighted sum of inputs plus an offset. If there is just one input $x$, then this is a straight line:\n", "\n", - "\\begin{equation}y=\\beta+\\omega x,\\end{equation}
\n", + "\\begin{equation}y=\\beta+\\omega x,\\end{equation} \n", "\n", "where $\\beta$ is the y-intercept of the linear and $\\omega$ is the slope of the line. When there are two inputs $x_{1}$ and $x_{2}$, then this becomes:\n", "\n", - "\\begin{equation}y=\\beta+\\omega_1 x_1 + \\omega_2 x_2.\\end{equation}

\n", + "\\begin{equation}y=\\beta+\\omega_1 x_1 + \\omega_2 x_2.\\end{equation} \n", "\n", "Any other functions are by definition **non-linear**.\n", "\n", @@ -104,6 +107,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "AedfvD9dxShZ" @@ -188,6 +192,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "i8tLwpls476R" @@ -195,15 +200,15 @@ "source": [ "Often we will want to compute many linear functions at the same time. For example, we might have three inputs, $x_1$, $x_2$, and $x_3$ and want to compute two linear functions giving $y_1$ and $y_2$. Of course, we could do this by just running each equation separately,

\n", "\n", - "\\begin{eqnarray}y_1 &=& \\beta_1 + \\omega_{11} x_1 + \\omega_{12} x_2 + \\omega_{13} x_3\\\\\n", + "\\begin{align}y_1 &=& \\beta_1 + \\omega_{11} x_1 + \\omega_{12} x_2 + \\omega_{13} x_3\\\\\n", "y_2 &=& \\beta_2 + \\omega_{21} x_1 + \\omega_{22} x_2 + \\omega_{23} x_3.\n", - "\\end{eqnarray}
\n", + "\\end{align}\n", "\n", "However, we can write it more compactly with vectors and matrices:\n", "\n", "\\begin{equation}\n", "\\begin{bmatrix} y_1\\\\ y_2 \\end{bmatrix} = \\begin{bmatrix}\\beta_{1}\\\\\\beta_{2}\\end{bmatrix}+ \\begin{bmatrix}\\omega_{11}&\\omega_{12}&\\omega_{13}\\\\\\omega_{21}&\\omega_{22}&\\omega_{23}\\end{bmatrix}\\begin{bmatrix}x_{1}\\\\x_{2}\\\\x_{3}\\end{bmatrix},\n", - "\\end{equation}
\n", + "\\end{equation}\n", "or\n", "\n", "\\begin{equation}\n", @@ -231,6 +236,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "fGzVJQ6N-mHJ" @@ -273,6 +279,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "3LGRoTMLU8ZU" @@ -286,6 +293,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "7Y5zdKtKZAB2" @@ -295,7 +303,7 @@ "\n", "Throughout the book, we'll be using some special functions (see Appendix B.1.3). The most important of these are the logarithm and exponential functions. Let's investigate their properties.\n", "\n", - "We'll start with the exponential function $y=\\mbox{exp}[x]=e^x$ which maps the real line $[-\\infty,+\\infty]$ to non-negative numbers $[0,+\\infty]$." + "We'll start with the exponential function $y=\\exp[x]=e^x$ which maps the real line $[-\\infty,+\\infty]$ to non-negative numbers $[0,+\\infty]$." ] }, { @@ -321,6 +329,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "XyrT8257IWCu" @@ -328,14 +337,15 @@ "source": [ "# Questions\n", "\n", - "1. What is $\\mbox{exp}[0]$? \n", - "2. What is $\\mbox{exp}[1]$?\n", - "3. What is $\\mbox{exp}[-\\infty]$?\n", - "4. What is $\\mbox{exp}[+\\infty]$?\n", + "1. What is $\\exp[0]$? \n", + "2. What is $\\exp[1]$?\n", + "3. What is $\\exp[-\\infty]$?\n", + "4. What is $\\exp[+\\infty]$?\n", "5. A function is convex if we can draw a straight line between any two points on the function, and this line always lies above the function. Similarly, a function is concave if a straight line between any two points always lies below the function. Is the exponential function convex or concave or neither?\n" ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "R6A4e5IxIWCu" @@ -367,6 +377,7 @@ ] }, { + "attachments": {}, "cell_type": "markdown", "metadata": { "id": "yYWrL5AXIWCv" @@ -374,20 +385,20 @@ "source": [ "# Questions\n", "\n", - "1. What is $\\mbox{log}[0]$? \n", - "2. What is $\\mbox{log}[1]$?\n", - "3. What is $\\mbox{log}[e]$?\n", - "4. What is $\\mbox{log}[\\exp[3]]$?\n", - "5. What is $\\mbox{exp}[\\log[4]]$?\n", - "6. What is $\\mbox{log}[-1]$?\n", + "1. What is $\\log[0]$? \n", + "2. What is $\\log[1]$?\n", + "3. What is $\\log[e]$?\n", + "4. What is $\\log[\\exp[3]]$?\n", + "5. What is $\\exp[\\log[4]]$?\n", + "6. What is $\\log[-1]$?\n", "7. Is the logarithm function concave or convex?\n" ] } ], "metadata": { "colab": { - "provenance": [], - "include_colab_link": true + "include_colab_link": true, + "provenance": [] }, "kernelspec": { "display_name": "Python 3 (ipykernel)",