Added content

Advanced/Linear Algebra 101/main.tex

@@ -9,7 +9,16 @@
 %\usepackage{lua-visual-debug}
 
 \usepackage{tikz-3dplot}
-\usetikzlibrary{quotes,angles}
+\usetikzlibrary{
+	quotes,
+	angles,
+	matrix,
+	decorations.pathreplacing,
+	calc,
+	positioning,
+	fit
+}
+\input{tikzset}
 
 \begin{document}
 
@@ -24,97 +33,10 @@
 
 	\input{parts/0 notation}
 	\input{parts/1 vectors}
-
-	\section{Dot Products}
-
-	\definition{}
-	We can also define the \textit{dot product} of two vectors.\footnotemark{} \\
-	The dot product maps two elements of $\mathbb{R}^n$ to one element of $\mathbb{R}$:
-
-	\footnotetext{
-		\textbf{Bonus content. Feel free to skip.}
-
-		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?
-
-		\vspace{2mm}
-
-		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.
-	}
-
-	$$
-		a \cdot b = \sum_{i = 1}^n a_ib_i = a_1b_1 + a_2b_2 + ... + a_nb_n
-	$$
-
-	\problem{}
-	Compute $[2, 3, 4, 1] \cdot [2, 4, 10, 12]$
-
-	\vfill
-
-	\problem{}
-	Show that the dot product is
-	\begin{itemize}
-		\item Commutative
-		\item Distributive
-		\item Homogeneic: $x(a \cdot b) = xa \cdot b = a \cdot xb$
-		\item Positive definite: $a \cdot a \geq 0$, with equality iff $a = 0$
-	\end{itemize}
+	\input{parts/2 dotprod}
+	\input{parts/3 matrices}
 
 
-	\vfill
-	\pagebreak
-
-
-
-
-	\problem{}
-	Say you have two vectors, $a$ and $b$. Show that $\langle a, b \rangle$ = $||a||~||b||\cos(\alpha)$ \\
-	\hint{What is $c$ in terms of $a$ and $b$?}
-	\hint{The law of cosines is $a^2 + b^2 - 2ab\cos(\alpha) = c^2$}
-	\hint{The length of $a$ is $||a||$}
-
-
-	\begin{center}
-	\begin{tikzpicture}[scale=1]
-
-		\draw[->]
-			(0,0) coordinate (o) -- node[above left] {$a$}
-			(1,2) coordinate (a)
-		;
-
-		\draw[->]
-			(o) -- node[below] {$b$}
-			(3,0.5) coordinate (b)
-		;
-
-		\draw[
-			draw = gray,
-			text = gray,
-			-
-		] (a) -- node[above] {$c$} (b);
-
-		\draw
-			pic[
-				"$\alpha$",
-				draw = orange,
-				text = orange,
-				<->,
-				angle eccentricity = 1.2,
-				angle radius = 1cm
-			]
-			{ angle = b--o--a }
-		;
-
-	\end{tikzpicture}
-	\end{center}
-
-	\vfill
-
-	\problem{}
-	If $a$ and $b$ are perpendicular, what must $\langle a, b \rangle$ be? Is the converse true?
-
-
-	\vfill
-	\pagebreak
 
 	\section{Bonus}