148 lines
2.7 KiB
TeX
Executable File
148 lines
2.7 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{lua-visual-debug}
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\usepackage{tikz-3dplot}
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\usetikzlibrary{quotes,angles}
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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 Algebra 101}
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{
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Prepared by Mark on \today \\
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}
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\input{parts/0 notation}
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\input{parts/1 vectors}
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\section{Dot Products}
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\definition{}
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We can also define the \textit{dot product} of two vectors.\footnotemark{} \\
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The dot product maps two elements of $\mathbb{R}^n$ to one element of $\mathbb{R}$:
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\footnotetext{
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\textbf{Bonus content. Feel free to skip.}
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Formally, we would say that the dot product is a map from $\mathbb{R}^n \times \mathbb{R}^n$ to $\mathbb{R}$. Why is this reasonable?
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\vspace{2mm}
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It's also worth noting that a function $f$ from $X$ to $Y$ can defined as a subset of $X \times Y$, where for all $x \in X$ there exists a unique $y \in Y$ so that $(x, y) \in f$. Try to make sense of this definition.
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}
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$$
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a \cdot b = \sum_{i = 1}^n a_ib_i = a_1b_1 + a_2b_2 + ... + a_nb_n
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$$
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\problem{}
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Compute $[2, 3, 4, 1] \cdot [2, 4, 10, 12]$
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\vfill
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\problem{}
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Show that the dot product is
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\begin{itemize}
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\item Commutative
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\item Distributive
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\item Homogeneic: $x(a \cdot b) = xa \cdot b = a \cdot xb$
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\item Positive definite: $a \cdot a \geq 0$, with equality iff $a = 0$
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\end{itemize}
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\vfill
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\pagebreak
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\problem{}
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Say you have two vectors, $a$ and $b$. Show that $\langle a, b \rangle$ = $||a||~||b||\cos(\alpha)$ \\
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\hint{What is $c$ in terms of $a$ and $b$?}
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\hint{The law of cosines is $a^2 + b^2 - 2ab\cos(\alpha) = c^2$}
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\hint{The length of $a$ is $||a||$}
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\begin{center}
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\begin{tikzpicture}[scale=1]
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\draw[->]
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(0,0) coordinate (o) -- node[above left] {$a$}
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(1,2) coordinate (a)
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;
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\draw[->]
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(o) -- node[below] {$b$}
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(3,0.5) coordinate (b)
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;
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\draw[
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draw = gray,
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text = gray,
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-
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] (a) -- node[above] {$c$} (b);
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\draw
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pic[
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"$\alpha$",
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draw = orange,
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text = orange,
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<->,
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angle eccentricity = 1.2,
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angle radius = 1cm
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]
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{ angle = b--o--a }
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;
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\end{tikzpicture}
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\end{center}
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\vfill
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\problem{}
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If $a$ and $b$ are perpendicular, what must $\langle a, b \rangle$ be? Is the converse true?
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\vfill
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\pagebreak
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\section{Bonus}
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\problem{}
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Show that the euclidean norm satisfies the triangle inequalty:
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$$
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||x+y|| \leq ||x|| + ||y||
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$$:
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\vfill
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\problem{}
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Show that the eucidean 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} |