88 lines
1.9 KiB
TeX
Executable File
88 lines
1.9 KiB
TeX
Executable File
% use [nosolutions] flag to hide solutions.
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% use [solutions] flag to show solutions.
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\documentclass[
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solutions,
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nowarning,
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%singlenumbering
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]{../../resources/ormc_handout}
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\usepackage{tikz}
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\usetikzlibrary{
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matrix,
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decorations.pathreplacing,
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calc,
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positioning,
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fit
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}
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% Let's give clarifications about the meaning of Z and R when we use them in the first problems.
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% 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".
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% Maybe we can add an example of a linear transformation from R^2 to R^2? Rotation? Scaling of y-axis?
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% Slow down, understand linear transformations fully.
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%\usepackage{lua-visual-debug}
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\renewcommand{\arraystretch}{1.2}
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\begin{document}
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\maketitle
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<Advanced 2>
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<Spring 2023>
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{Linear Maps}
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{
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Prepared by Mark on \today \\
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}
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\input{parts/0 fields}
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\input{parts/1 spaces}
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\input{parts/2 linear}
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\input{parts/3 matrices}
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\section{Norms}
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\definition{}
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If $V$ is a vector space, a \textit{norm} in $V$ is a function $V \to \mathbb{R}^+$ that satisfies the following properties, \\
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Where $x, y \in V$ and $c \in F$:
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\begin{itemize}
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\item Absolute Homogeneity: $||cx|| = |c|~||x||$
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\item Positive-Definite: $||x|| \geq 0$ with equality iff $x = 0$.
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\item Triangle Inequalty: $||x+y|| \leq ||x|| + ||y||$
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\end{itemize}
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\problem{}
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Show that the \textit{euclidian norm} defined by $||~[a, b]~|| = \sqrt{a^2 + b^2}$ is a norm on $\mathbb{R}^2$
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\vfill
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\problem{}
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Show that in any vector space with an inner product, the \textit{induced norm} $||x|| = \sqrt{\langle x, x \rangle}$ is a norm.
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\vfill
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\problem{}
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Show that every norm satisfies the reverse triangle inequality:
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$$
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||x - y|| \geq |~||x|| - ||y||~|
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$$
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\vfill
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\problem{}
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Prove the Cauchy-Schwartz inequality:
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$$
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||\langle x, y \rangle|| = ||x||~||y||
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$$
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\vfill
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\end{document} |