% 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}