Linear Algebra edits
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@ -9,6 +9,7 @@
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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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@ -21,131 +22,120 @@
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}
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\section{Notation and Terminology}
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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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\begin{itemize}
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\item $\mathbb{R}$ is the set of all real numbers.
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\item $\mathbb{R}^+$ is the set of positive real numbers. Zero is not positive.
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\item $\mathbb{R}^+_0$ is the set of positive real numbers and zero
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\end{itemize}
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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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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.
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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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\definition{}
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Consider two sets $A$ and $B$. The set $A \times B$ consists of all tuples $(a, b)$ where $a \in A$ and $b \in B$. \\
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For example, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit), (1, \star), (2,\heartsuit), (2, \star), (3,\heartsuit), (3, \star)\}$ \\
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\vspace{2mm}
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\vspace{4mm}
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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 $A \times B$:
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\begin{center}
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\begin{tikzpicture}[
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scale=1,
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bullet/.style={circle,inner sep=1.5pt,fill}
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]
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\draw[->] (-0.2,0) -- (4,0) node[right]{$A$};
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\draw[->] (0,-0.2) -- (0,3) node[above]{$B$};
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\draw (1,0.1) -- ++ (0,-0.2) node[below]{$1$};
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\draw (2,0.1) -- ++ (0,-0.2) node[below]{$2$};
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\draw (3,0.1) -- ++ (0,-0.2) node[below]{$3$};
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\draw (0.1, 1) -- ++ (-0.2, 0) node[left]{$\heartsuit$};
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\draw (0.1, 2) -- ++ (-0.2, 0) node[left]{$\star$};
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\node[bullet] at (1, 1){};
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\node[bullet] at (2, 1) {};
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\node[bullet] at (3, 1) {};
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\node[bullet] at (1, 2) {};
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\node[bullet] at (2, 2) {};
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\node[bullet] at (3, 2) {};
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\end{tikzpicture}
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\end{center}
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\problem{}
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Let $A = \{0, 1\} \times \{0, 1\}$. \\
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Let $B = \{ a, b\}$ \\
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What is $A \times B$?
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\vfill
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\problem{}
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What is $\mathbb{R} \times \mathbb{R}$? \\
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\hint{Use the \say{perpendicular} analogy}
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\vfill
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\pagebreak
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\definition{}
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$\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \\
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In english, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \\
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\vspace{4mm}
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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.}}
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Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4 a_5)$, where $a_n \in \mathbb{R}$. \\
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$\mathbb{R}^1$ and $\mathbb{R}$ are identical.
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\vspace{4mm}
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Intuitively, $\mathbb{R}^2$ forms a two-dimensional plane, and $\mathbb{R}^3$ forms a three-dimensional space. \\
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$\mathbb{R}^n$ is hard to visualize when $n \geq 4$, but you are welcome to try.
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\problem{}
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Convince yourself that $\mathbb{R} \times \mathbb{R}$ is $\mathbb{R}^2$. \\
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What is $\mathbb{R}^2 \times \mathbb{R}$?
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\vfill
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\section{Vectors}
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\definition{}
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Elements of $\mathbb{R}^n$ are often called \textit{vectors}. \\
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As you already know, we have a few operations on vectors:
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\begin{itemize}
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\item Vector addition: $[a_1, a_2] + [b_1, b_2] = [a_1+b_1, a_2+b_2]$
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\item Scalar multiplication: $x \times [a_1, a_2] = [xa_1, xa_2]$.
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\end{itemize}
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\problem{}
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Compute the following, or explain why you can't:
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\begin{itemize}
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\item $[1, 2, 3] + [1, 3, 4]$
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\item $4 \times [5, 2, 4]$
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\item $a + b$, where $a \in \mathbb{R}
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^5$ and $b \in \mathbb{R}^7$
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\end{itemize}
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\vfill
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\pagebreak
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\definition{}
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We can also define the \textit{dot product} of two vectors. \\
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The dot product maps a pair of elements from $\mathbb{R}^n$ to $\mathbb{R}$:
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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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% pictures
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% addition, scalar multiplication
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% dot product
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% transformations
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% linearity
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% matrices
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% norms
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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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\pagebreak
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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\begin{tikzpicture}[scale=1]
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\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
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\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
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\draw[->] (0,0) -- (1,2) node[right]{};
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\draw[->] (0,0) -- (3,0.5) node[above]{};
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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\begin{tikzpicture}[scale=1]
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\draw[dashed,->] (-0.5,0) -- (4,0) node[right]{};
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\draw[dashed,->] (0,-0.5) -- (0,3) node[above]{};
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\draw[->] (0,0) -- (3,1) node[right]{};
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\draw[->] (0,0) -- (3,0.5) node[above]{};
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\end{tikzpicture}
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\end{center}
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\end{minipage}
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87
Advanced/Linear Algebra 101/parts/0 notation.tex
Executable file
87
Advanced/Linear Algebra 101/parts/0 notation.tex
Executable file
@ -0,0 +1,87 @@
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\section{Notation and Terminology}
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\definition{}
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\begin{itemize}
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\item $\mathbb{R}$ is the set of all real numbers.
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\item $\mathbb{R}^+$ is the set of positive real numbers. Zero is not positive.
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\item $\mathbb{R}^+_0$ is the set of positive real numbers and zero.
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\end{itemize}
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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.
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\definition{}
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Consider two sets $A$ and $B$. The set $A \times B$ consists of all tuples $(a, b)$ where $a \in A$ and $b \in B$. \\
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For example, $\{1, 2, 3\} \times \{\heartsuit, \star\} = \{(1,\heartsuit), (1, \star), (2,\heartsuit), (2, \star), (3,\heartsuit), (3, \star)\}$ \\
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This is called the \textit{cartesian product}.
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\vspace{4mm}
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You can think of this as placing the two sets \say{perpendicular} to one another:
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\begin{center}
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\begin{tikzpicture}[
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scale=1,
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bullet/.style={circle,inner sep=1.5pt,fill}
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]
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\draw[->] (-0.2,0) -- (4,0) node[right]{$A$};
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\draw[->] (0,-0.2) -- (0,3) node[above]{$B$};
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\draw (1,0.1) -- ++ (0,-0.2) node[below]{$1$};
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\draw (2,0.1) -- ++ (0,-0.2) node[below]{$2$};
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\draw (3,0.1) -- ++ (0,-0.2) node[below]{$3$};
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\draw (0.1, 1) -- ++ (-0.2, 0) node[left]{$\heartsuit$};
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\draw (0.1, 2) -- ++ (-0.2, 0) node[left]{$\star$};
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\node[bullet] at (1, 1){};
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\node[bullet] at (2, 1) {};
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\node[bullet] at (3, 1) {};
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\node[bullet] at (1, 2) {};
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\node[bullet] at (2, 2) {};
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\node[bullet] at (3, 2) {};
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\draw[rounded corners] (0.5, 0.5) rectangle (3.5, 2.5) {};
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\node[above] at (2, 2.5) {$A \times B$};
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\end{tikzpicture}
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\end{center}
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\problem{}
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Let $A = \{0, 1\} \times \{0, 1\}$ \\
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Let $B = \{ a, b\}$ \\
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What is $A \times B$?
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\vfill
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\problem{}
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What is $\mathbb{R} \times \mathbb{R}$? \\
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\hint{Use the \say{perpendicular} analogy}
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\vfill
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\pagebreak
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\definition{}
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$\mathbb{R}^n$ is the set of $n$-tuples of real numbers. \\
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In english, this means that an element of $\mathbb{R}^n$ is a list of $n$ real numbers: \\
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\vspace{4mm}
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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.}}
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Elements of $\mathbb{R}^5$ look like $(a_1, a_2, a_3, a_4 a_5)$, where $a_n \in \mathbb{R}$. \\
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$\mathbb{R}^1$ and $\mathbb{R}$ are identical.
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\vspace{4mm}
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Intuitively, $\mathbb{R}^2$ forms a two-dimensional plane, and $\mathbb{R}^3$ forms a three-dimensional space. \\
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$\mathbb{R}^n$ is hard to visualize when $n \geq 4$, but you are welcome to try.
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\problem{}
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Convince yourself that $\mathbb{R} \times \mathbb{R}$ is $\mathbb{R}^2$. \\
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What is $\mathbb{R}^2 \times \mathbb{R}$?
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\vfill
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\pagebreak
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81
Advanced/Linear Algebra 101/parts/1 vectors.tex
Executable file
81
Advanced/Linear Algebra 101/parts/1 vectors.tex
Executable file
@ -0,0 +1,81 @@
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\section{Vectors}
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\definition{}
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Elements of $\mathbb{R}^n$ are often called \textit{vectors}. \\
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As you may already know, we have a few operations on vectors:
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\begin{itemize}
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\item Vector addition: $[a_1, a_2] + [b_1, b_2] = [a_1+b_1, a_2+b_2]$
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\item Scalar multiplication: $x \times [a_1, a_2] = [xa_1, xa_2]$.
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\end{itemize}
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\note{
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The above examples are for $\mathbb{R}^2$, and each vector thus has two components. \\
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These operations are similar for all other $n$.
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}
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\problem{}
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Compute the following or explain why you can't:
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\begin{itemize}
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\item $[1, 2, 3] - [1, 3, 4]$ \note{Subtraction works just like addition.}
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\item $4 \times [5, 2, 4]$
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\item $a + b$, where $a \in \mathbb{R}
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^5$ and $b \in \mathbb{R}^7$
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\end{itemize}
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\vfill
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\problem{}
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Consider $(2, -1)$ and $(3, 1)$ in $\mathbb{R}^2$. \\
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Can you develop geometric intuition for their sum and difference?
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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[below left] {$(1, 2)$}
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(2, -1) coordinate (a)
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;
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\draw[->]
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(a) -- node[below right] {$(3, 1)$}
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(5, 0) 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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]
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(o) -- node[above] {$??$}
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(b) coordinate (s)
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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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\pagebreak
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\definition{Euclidean Norm}
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In general, a \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
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Usually, one thinks of a norm as a \say{length metric} on a vector space. \\
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The norm of a vector $v$ is written $||v||$. \\
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\vspace{2mm}
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We usually use the \textit{euclidean norm} when we work in $\mathbb{R}^n$. \\
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If $v \in \mathbb{R}^n$, the euclidean norm is defined as follows:
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$$
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||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
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$$
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This is simply an application of the pythagorean theorem.
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\problem{}
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Compute the euclidean norm of
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\begin{itemize}
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\item $[2, 3]$
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\item $[-2, 1, -4, 2]$
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\end{itemize}
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\vfill
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\pagebreak
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