Started basic LA handout

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
+		<Advanced 2>
+		<Spring 2023>
+		{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}