From eaa5978dc92c56f24b5ff0f77dcac40aff6a4233 Mon Sep 17 00:00:00 2001 From: Mark Date: Wed, 5 Apr 2023 15:31:36 -0700 Subject: [PATCH] Started basic LA handout --- Advanced/Linear Algebra 101/main.tex | 153 +++++++++++++++++++++++++++ 1 file changed, 153 insertions(+) create mode 100755 Advanced/Linear Algebra 101/main.tex diff --git a/Advanced/Linear Algebra 101/main.tex b/Advanced/Linear Algebra 101/main.tex new file mode 100755 index 0000000..719d081 --- /dev/null +++ b/Advanced/Linear Algebra 101/main.tex @@ -0,0 +1,153 @@ +% use [nosolutions] flag to hide solutions. +% use [solutions] flag to show solutions. +\documentclass[ + solutions, + nowarning, + %singlenumbering +]{../../resources/ormc_handout} + +%\usepackage{lua-visual-debug} + +\usepackage{tikz-3dplot} + +\begin{document} + + \maketitle + + + {Linear Algebra 101} + { + Prepared by Mark on \today \\ + } + + + \section{Notation and Terminology} + + \definition{} + \begin{itemize} + \item $\mathbb{R}$ is the set of all real numbers. + \item $\mathbb{R}^+$ is the set of positive real numbers. Zero is not positive. + \item $\mathbb{R}^+_0$ is the set of positive real numbers and zero + \end{itemize} + + Mathematicians are often inconsistent with their notation. Depending on the author, their mood, and the phase of the moon, $\mathbb{R}^+$ may or may not include zero. I will use the definitions above. + + + \definition{} + Consider two sets $A$ and $B$. The set $A \times B$ consists of all tuples $(a, b)$ where $a \in A$ and $b \in B$. \\ + For example, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit), (1, \star), (2,\heartsuit), (2, \star), (3,\heartsuit), (3, \star)\}$ \\ + + \vspace{4mm} + + You can think of this as placing the two sets \say{perpendicular} to one another. In the image below, each dot corresponds to an element of $A \times B$: + + \begin{center} + \begin{tikzpicture}[ + scale=1, + bullet/.style={circle,inner sep=1.5pt,fill} + ] + \draw[->] (-0.2,0) -- (4,0) node[right]{$A$}; + \draw[->] (0,-0.2) -- (0,3) node[above]{$B$}; + + \draw (1,0.1) -- ++ (0,-0.2) node[below]{$1$}; + \draw (2,0.1) -- ++ (0,-0.2) node[below]{$2$}; + \draw (3,0.1) -- ++ (0,-0.2) node[below]{$3$}; + + \draw (0.1, 1) -- ++ (-0.2, 0) node[left]{$\heartsuit$}; + \draw (0.1, 2) -- ++ (-0.2, 0) node[left]{$\star$}; + + \node[bullet] at (1, 1){}; + \node[bullet] at (2, 1) {}; + \node[bullet] at (3, 1) {}; + \node[bullet] at (1, 2) {}; + \node[bullet] at (2, 2) {}; + \node[bullet] at (3, 2) {}; + + \end{tikzpicture} + \end{center} + + \problem{} + Let $A = \{0, 1\} \times \{0, 1\}$. \\ + Let $B = \{ a, b\}$ \\ + What is $A \times B$? + + \vfill + + \problem{} + What is $\mathbb{R} \times \mathbb{R}$? \\ + \hint{Use the \say{perpendicular} analogy} + + \vfill + \pagebreak + + \definition{} + $\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \\ + In english, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \\ + + \vspace{4mm} + + Elements of $\mathbb{R}^2$ look like $(a, b)$, where $a, b \in \mathbb{R}$. \hfill \note{\textit{Note:} $\mathbb{R}^2$ is pronounced \say{arrgh-two.}} + Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4 a_5)$, where $a_n \in \mathbb{R}$. \\ + + $\mathbb{R}^1$ and $\mathbb{R}$ are identical. + + \vspace{4mm} + + Intuitively, $\mathbb{R}^2$ forms a two-dimensional plane, and $\mathbb{R}^3$ forms a three-dimensional space. \\ + $\mathbb{R}^n$ is hard to visualize when $n \geq 4$, but you are welcome to try. + + \problem{} + Convince yourself that $\mathbb{R} \times \mathbb{R}$ is $\mathbb{R}^2$. \\ + What is $\mathbb{R}^2 \times \mathbb{R}$? + + + \vfill + + + \section{Vectors} + + \definition{} + Elements of $\mathbb{R}^n$ are often called \textit{vectors}. \\ + As you already know, we have a few operations on vectors: + \begin{itemize} + \item Vector addition: $[a_1, a_2] + [b_1, b_2] = [a_1+b_1, a_2+b_2]$ + \item Scalar multiplication: $x \times [a_1, a_2] = [xa_1, xa_2]$. + \end{itemize} + + + \problem{} + Compute the following, or explain why you can't: + \begin{itemize} + \item $[1, 2, 3] + [1, 3, 4]$ + \item $4 \times [5, 2, 4]$ + \item $a + b$, where $a \in \mathbb{R} +^5$ and $b \in \mathbb{R}^7$ + \end{itemize} + + \vfill + \pagebreak + + \definition{} + We can also define the \textit{dot product} of two vectors. \\ + The dot product maps a pair of elements from $\mathbb{R}^n$ to $\mathbb{R}$: + + $$ + a \cdot b = \sum_{i = 1}^n a_ib_i = a_1b_1 + a_2b_2 + ... + a_nb_n + $$ + + \problem{} + + + + % pictures + % addition, scalar multiplication + % dot product + % transformations + % linearity + % matrices + % norms + + + + +\end{document} \ No newline at end of file