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udlbook
2024-01-02 11:58:12 -05:00
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51
Notebooks/Chap01/1_1_BackgroundMathematics.ipynb
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"<a href=\"https://colab.research.google.com/github/udlbook/udlbook/blob/main/Notebooks/Chap01/1_1_BackgroundMathematics.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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"**Linear functions**<br> 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} <br>\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} <br><br>\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",
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"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,<br><br>\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}<br>\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}<br>\n",
"\\end{equation}\n",
"or\n",
"\n",
"\\begin{equation}\n",
@@ -231,6 +236,7 @@
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"\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 $[0,+\\infty]$."
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@@ -328,14 +337,15 @@
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"# 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"
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@@ -374,20 +385,20 @@
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"# 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"
]
}
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