Linear Algebra edits

Advanced/Linear Algebra 101/main.tex

@@ -9,6 +9,7 @@
 %\usepackage{lua-visual-debug}
 
 \usepackage{tikz-3dplot}
+\usetikzlibrary{quotes,angles}
 
 \begin{document}
 
@@ -21,131 +22,120 @@
 		}
 
 
-	\section{Notation and Terminology}
+	\input{parts/0 notation}
+	\input{parts/1 vectors}
+
+	\section{Dot Products}
 
 	\definition{}
-	\begin{itemize}
+	We can also define the \textit{dot product} of two vectors.\footnotemark{} \\
-		\item $\mathbb{R}$ is the set of all real numbers.
+	The dot product maps two elements of $\mathbb{R}^n$ to one element of $\mathbb{R}$:
-		\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.
+	\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?
 
-	\definition{}
+		\vspace{2mm}
-	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}
+		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.
-
+	}
-	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{}
+	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}
+
+
+	\vfill
+	\pagebreak
 
 
 
-	% pictures
+
-	% addition, scalar multiplication
+	\problem{}
-	% dot product
+	Say you have two vectors, $a$ and $b$. Show that $\langle a, b \rangle$ = $||a||~||b||\cos(\alpha)$ \\
-		% transformations
+	\hint{What is $c$ in terms of $a$ and $b$?}
-			% linearity
+	\hint{The law of cosines is $a^2 + b^2 - 2ab\cos(\alpha) = c^2$}
-			% matrices
+	\hint{The length of $a$ is $||a||$}
-		% norms
+
+
+	\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
+	\pagebreak
+
+	\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
+	\begin{center}
+	\begin{tikzpicture}[scale=1]
+		\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
+		\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
+
+		\draw[->] (0,0) -- (1,2) node[right]{};
+		\draw[->] (0,0) -- (3,0.5) node[above]{};
+	\end{tikzpicture}
+	\end{center}
+	\end{minipage}
+	\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
+	\begin{center}
+	\begin{tikzpicture}[scale=1]
+		\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
+		\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
+
+		\draw[->] (0,0) -- (3,1) node[right]{};
+		\draw[->] (0,0) -- (3,0.5) node[above]{};
+	\end{tikzpicture}
+	\end{center}
+	\end{minipage}
+
 
 
 

Advanced/Linear Algebra 101/parts/0 notation.tex

@@ -0,0 +1,87 @@
+\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, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit), (1, \star), (2,\heartsuit), (2, \star), (3,\heartsuit), (3, \star)\}$ \\
+This is called the \textit{cartesian product}.
+
+\vspace{4mm}
+
+You can think of this as placing the two sets \say{perpendicular} to one another:
+
+\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) {};
+
+
+	\draw[rounded corners] (0.5, 0.5) rectangle (3.5, 2.5) {};
+	\node[above] at (2, 2.5) {$A \times B$};
+
+\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
+\pagebreak

Advanced/Linear Algebra 101/parts/1 vectors.tex

@@ -0,0 +1,81 @@
+\section{Vectors}
+
+\definition{}
+Elements of $\mathbb{R}^n$ are often called \textit{vectors}. \\
+As you may 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}
+\note{
+	The above examples are for $\mathbb{R}^2$, and each vector thus has two components. \\
+	These operations are similar for all other $n$.
+}
+
+\problem{}
+Compute the following or explain why you can't:
+\begin{itemize}
+	\item $[1, 2, 3] - [1, 3, 4]$ \note{Subtraction works just like addition.}
+	\item $4 \times [5, 2, 4]$
+	\item $a + b$, where $a \in \mathbb{R}
+^5$ and $b \in \mathbb{R}^7$
+\end{itemize}
+
+\vfill
+
+\problem{}
+Consider $(2, -1)$ and $(3, 1)$ in $\mathbb{R}^2$. \\
+Can you develop geometric intuition for their sum and difference?
+
+\begin{center}
+	\begin{tikzpicture}[scale=1]
+
+		\draw[->]
+			(0,0) coordinate (o) -- node[below left] {$(1, 2)$}
+			(2, -1) coordinate (a)
+		;
+
+		\draw[->]
+			(a) -- node[below right] {$(3, 1)$}
+			(5, 0) coordinate (b)
+		;
+
+		\draw[
+			draw = gray,
+			text = gray,
+			->
+		]
+			(o) -- node[above] {$??$}
+			(b) coordinate (s)
+		;
+
+	\end{tikzpicture}
+	\end{center}
+
+
+\vfill
+\pagebreak
+
+\definition{Euclidean Norm}
+In general, a \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
+Usually, one thinks of a norm as a \say{length metric} on a vector space. \\
+The norm of a vector $v$ is written $||v||$. \\
+
+\vspace{2mm}
+
+We usually use the \textit{euclidean norm} when we work in $\mathbb{R}^n$. \\
+If $v \in \mathbb{R}^n$, the euclidean norm is defined as follows:
+$$
+	||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
+$$
+This is simply an application of the pythagorean theorem.
+
+\problem{}
+Compute the euclidean norm of
+\begin{itemize}
+	\item $[2, 3]$
+	\item $[-2, 1, -4, 2]$
+\end{itemize}
+
+\vfill
+\pagebreak