% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
	solutions,
	nowarning,
	%singlenumbering
]{../../resources/ormc_handout}

\usepackage{tikz}
\usetikzlibrary{
	matrix,
	decorations.pathreplacing,
	calc,
	positioning,
	fit
}


% Let's give clarifications about the meaning of Z and R when we use them in the first problems.

% Definitely define $R^n$ before using. Optionally you may add a problem "convince yourself that $R^2$ is a plane and $R^3$ is a 3-d space".

% Maybe we can add an example of a linear transformation from R^2 to R^2? Rotation? Scaling of y-axis?

% Slow down, understand linear transformations fully.




%\usepackage{lua-visual-debug}
\renewcommand{\arraystretch}{1.2}
\begin{document}

	\maketitle
		<Advanced 2>
		<Spring 2023>
		{Linear Maps}
		{
			Prepared by Mark on \today \\
		}

	\input{parts/0 fields}
	\input{parts/1 spaces}
	\input{parts/2 linear}
	\input{parts/3 matrices}


	\section{Norms}

	\definition{}
	If $V$ is a vector space, a \textit{norm} in $V$ is a function $V \to \mathbb{R}^+$ that satisfies the following properties, \\
	Where $x, y \in V$ and $c \in F$:

	\begin{itemize}
		\item Absolute Homogeneity: $||cx|| = |c|~||x||$
		\item Positive-Definite: $||x|| \geq 0$ with equality iff $x = 0$.
		\item Triangle Inequalty: $||x+y|| \leq ||x|| + ||y||$
	\end{itemize}

	\problem{}
	Show that the \textit{euclidian norm} defined by $||~[a, b]~|| = \sqrt{a^2 + b^2}$ is a norm on $\mathbb{R}^2$

	\vfill

	\problem{}
	Show that in any vector space with an inner product, the \textit{induced norm} $||x|| = \sqrt{\langle x, x \rangle}$ is a norm.

	\vfill

	\problem{}
	Show that every norm satisfies the reverse triangle inequality:

	$$
		||x - y|| \geq |~||x|| - ||y||~|
	$$

	\vfill

	\problem{}
	Prove the Cauchy-Schwartz inequality:

	$$
		||\langle x, y \rangle|| = ||x||~||y||
	$$

	\vfill

\end{document}