Added linear maps handout
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Advanced/Linear Maps/main.tex
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50
Advanced/Linear Maps/main.tex
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% 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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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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%\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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\section{Fields and Vector Spaces}
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\input{parts/0 fields}
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\input{parts/1 spaces}
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\input{parts/2 linearity}
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\input{parts/3 matrices}
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\section{Bonus}
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\problem{}
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Is the set of all linear maps a vector space?
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\vfill
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\definition{}
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Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space.
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\vfill
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\end{document}
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47
Advanced/Linear Maps/parts/0 fields.tex
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47
Advanced/Linear Maps/parts/0 fields.tex
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\definition{Fields and Field Axioms}
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A \textit{field} $\mathbb{F}$ consists of a set $A$ and two operations $+$ and $\times$. \\
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As usual, we may abbreviate $a \times b$ as $ab$. \\
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The following axioms must be satisfied for any $a, b, c \in \mathbb{F}$:
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\vspace{1mm}
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\begin{center}
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% @{} supresses the space between columns.
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% @{=} makes = a column seperator.
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\begin{tabular}{l | r@{=}l | r@{=}l}
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\hline
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\multicolumn{1}{|c|}{Name} &
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\multicolumn{2}{c}{$+$} &
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\multicolumn{2}{|c|}{$\times$} \\
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\hline
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Closure & \multicolumn{2}{c|}{$a+b \in \mathbb{F}$} & \multicolumn{2}{c}{$ab \in \mathbb{F}$} \\
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Associativity & $(a+b)+c~$&$~a+b+c$ & $(ab)c~$&$~a(bc)$ \\
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Commutativity & $a+b~$&$~b+a$ & $ab~$&$~ba$ \\
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Distributivity & $a(b+c)~$&$~ab + ac$ & \multicolumn{2}{}{} \\
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Identity & $a+0~$&$~a$ & $1 \times a~$&$~a$ \\
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Inverses & $a + (-a)~$&$~0$ & $a \times a^{-1}~$&$~1$
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\end{tabular}
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\end{center}
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\problem{}
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Show that all fields are groups. \\
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Convince yourself that not all groups are fields.
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\vfill
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\problem{}
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Is $\mathbb{Z}$ a field under our usual definitions of $+$ and $\times$? \\
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Which axioms does it satisfy, and which does it violate?
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\vfill
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\problem{}
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Verify that $\mathbb{R}$ is a field.
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\vfill
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\generic{Remark:}
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We won't worry too much about fields this week. They simply provide a foundation for \textit{spaces}. \\
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As such, you may assume that we are working in $\mathbb{R}$ for the rest of this handout.
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\pagebreak
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92
Advanced/Linear Maps/parts/1 spaces.tex
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92
Advanced/Linear Maps/parts/1 spaces.tex
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\definition{Vector Spaces}
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A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
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\begin{itemize}[itemsep = 2mm]
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\item A set $V$, the elements of which are called \textit{vectors}
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\item An operation called \textit{vector addition}, denoted $+$ \\
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Vector addition operates on two elements of $V$. \\
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\item An operation called \textit{scalar multilplication}, denoted $\times$ \\
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Scalar multiplication multiplies an element of $V$ by an element of $\mathbb{F}$. \\
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Any element of $\mathbb{F}$ is called a \textit{scalar}.
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\end{itemize}
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\vspace{2mm}
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\textbf{Note:}
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The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\
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Be careful, since \textit{these are different operations!} \\
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Make sure you're aware of the context of each $+$ and $\times$ as you work through this handout.
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\vspace{5mm}
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Vector addition and multiplication must have the following properties. \\
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Note that $x, y, z \in V$ and $a, b\in \mathbb{F}$.
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\vspace{2mm}
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% [t] and \vspace{0pt} ensure alignment at top
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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\begin{tabular}{l | r@{=}l }
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\hline
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\multicolumn{3}{|c|}{Properties of vector addition} \\
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\hline
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Closure & \multicolumn{2}{c}{$x+y \in V$} \\
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Associativity & $(x+y)+z~$&$~x+y+z$ \\
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Commutativity & $x+y~$&$~y+x$ \\
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Distributivity & $x(y+z)~$&$~xy + xz$ \\
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Identity & $x+0~$&$~x$ \\
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Inverse & $x + (-x)~$&$~0$
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\end{tabular}
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\end{center}
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\end{minipage}%
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\hfill%
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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\begin{tabular}{l | r@{=}l }
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\hline
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\multicolumn{3}{|c|}{Properties of vector multiplication} \\
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\hline
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Closure & \multicolumn{2}{c}{$ax \in V$} \\
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Distributivity & $a(x+y)~$&$~ax+ay$ \\
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& $(a+b)x~$&$~ax+bx$ \\
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Compatibility$^*$ & $(ab)x~$&$~x(ba)$ \\
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Identity & $a+0~$&$~a$
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\end{tabular}
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\end{center}
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\end{minipage}
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\vspace{5mm}
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\definition{}
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There is a good chance you are familiar with basic vector arithmetic. \\
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Here's a quick review:
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\begin{itemize}
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\item Scalar multiplication is done elementwise: $3 \times [a, b, c] = [3a, 3b, 3c]$.
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\item Vector addition is similar: $[a, b, c] + [1, 2, 3] = [a+1,~b+2,~c+3]$.
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\item Vector addition is not valid for vectors of different sizes.
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\end{itemize}
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\definition{}
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We usually use the \textit{dot product} as our vector product. It is defined as follows. \\
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Given two vectors $a, b \in \mathbb{R}^n$, the dot product $a \cdot b$ is $\sum_1^n a_ib_i$.
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\vspace{2mm}
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In other words, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
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$$
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a \cdot b = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
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$$
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As you may expect, the dot product $ab$ is valid iff $a$ and $b$ are the same size.
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\problem{}
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Show that the dot product satisfies the properties of a vector product listed above. \\
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Conclude that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$.
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\vfill
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\pagebreak
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42
Advanced/Linear Maps/parts/2 linearity.tex
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42
Advanced/Linear Maps/parts/2 linearity.tex
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\section{Linearity}
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\definition{}
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A \textit{function} or \textit{map} $f$ from a set $A$ to a set $B$ is a rule that assigns an element of $B$ to each element of $A$. We write this as $f: A \to B$.
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\definition{This one is worth remembering}
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Let $V$ and $U$ be vector spaces, and let $f: V \to U$ be a map from $V$ to $U$. \\
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We say $f$ is \textit{linear} if it satisfies the following for any $v \in V$, $u \in U$, $a \in \mathbb{R}$:
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\begin{itemize}
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\item $f(u + v) = f(u) + f(v)$
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\item $f(au) = af(u)$
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\end{itemize}
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In other words, $f$ is linear if it is \say{closed} under addition and scalar multiplication.
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\problem{}
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It is often convenient to combine the two conditions above into one. \\
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Show that $f(au + v) = af(u) + f(v)$ iff $f$ is linear.
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\vfill
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\problem{}
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Is $f(x) = mx + b$ a linear map on $\mathbb{R}$?
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\vfill
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\problem{}
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In general, what does a linear map in $\mathbb{R}^n$ look like?
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\vfill
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\problem{}
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Is $\text{median}(v): \mathbb{R}^n \to \mathbb{R}$ a linear map on $\mathbb{R}^n$?
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\vfill
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\problem{}
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Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$?
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\vfill
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\pagebreak
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198
Advanced/Linear Maps/parts/3 matrices.tex
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198
Advanced/Linear Maps/parts/3 matrices.tex
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\section{Matrices}
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\definition{}
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A \textit{matrix} is a two-dimensional array of numbers: \\
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$$
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A =
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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$$
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The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix.
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\problem{}
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Draw a $3 \times 2$ matrix.
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\vfill
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\definition{Matrices as Transformations}
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We can define the \say{product\footnotemark{}} of a matrix $A$ and a vector $v$:
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\footnotetext{This is an uncommon word to use in this context. You will soon see why.}
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$$
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Av =
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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a \\ b \\ c
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\end{bmatrix}
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=
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\begin{bmatrix}
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1a + 2b + 3c \\
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4a + 5b + 6c
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\end{bmatrix}
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$$
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Look closely. Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
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$$
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Av =
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\begin{bmatrix}
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\text{---} a_1 \text{---} \\
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\text{---} a_2 \text{---}
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\end{bmatrix}
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\begin{bmatrix}
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| \\
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v \\
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| \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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r_1v \\
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r_2v
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\end{bmatrix}
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$$
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Naturally, a vector can only be multiplied by a matrix if the number of rows in the vector equals the number of columns in the matrix.
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\problem{}
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Compute the following \say{product}:
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$$
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\begin{bmatrix}
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2 & 9 \\
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7 & 5 \\
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3 & 4
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\end{bmatrix}
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\begin{bmatrix}
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5 \\ 3
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\end{bmatrix}
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$$
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\vfill
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\pagebreak
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\generic{Remark:}
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It is a bit more interesting to think of matrix-vector multiplication in the following way: \\
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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The problem:
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\vspace{2mm}
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$$
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\begin{bmatrix}
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2 & 9 \\
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7 & 5 \\
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3 & 4
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\end{bmatrix}
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\begin{bmatrix}
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5 \\ 3
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\end{bmatrix}
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=
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\begin{bmatrix}
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37 \\ 50 \\ 27
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\end{bmatrix}
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$$
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\end{center}
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\end{minipage}%
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\hfill
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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Top-input, right-output:
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\vspace{2mm}
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\begin{tikzpicture}[>=stealth,thick,baseline]
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\matrix [
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matrix of math nodes,
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left delimiter={[},
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right delimiter={]}
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] (A) {
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2 & 9 \\
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7 & 5 \\
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3 & 4 \\
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};
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\node[
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fit=(A-1-1)(A-1-1),
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inner xsep=0mm,inner ysep=3mm,
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label=above:5
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] (L) {};
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\draw[->, gray] (L.north) -- ([yshift=0mm]A-1-1.north);
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\node[
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fit=(A-1-2)(A-1-2),
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inner xsep=0mm,inner ysep=3mm,
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label=above:3
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] (R) {};
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\draw[->, gray] (R.north) -- ([yshift=0mm]A-1-2.north);
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\node[
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fit=(A-1-2)(A-1-2),
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inner xsep=8mm,inner ysep=0mm,
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label=right:{$10 + 27 = 37$}
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](Y) {};
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\draw[->, gray] ([xshift=3mm]A-1-2.east) -- (Y);
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\node[
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fit=(A-2-2)(A-2-2),
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inner xsep=8mm,inner ysep=0mm,
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label=right:{$35 + 15 = 50$}
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](H) {};
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\draw[->, gray] ([xshift=3mm]A-2-2.east) -- (H);
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\node[
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fit=(A-3-2)(A-3-2),
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inner xsep=8mm,inner ysep=0mm,
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label=right:{$15 + 12 = 27$}
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](N) {};
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\draw[->, gray] ([xshift=3mm]A-3-2.east) -- (N);
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\end{tikzpicture}
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\end{center}
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\end{minipage}%
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\vspace{2mm}
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Be aware that this is only a model for intuition. \\
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Make sure you understand the dot product definition on the previous page.
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\vspace{5mm}
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\theorem{}<thebigtheorem>
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Any linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as an $n \times m$ matrix. \\
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Conversely, every $n \times m$ matrix represents a liner map $T: \mathbb{R}^n \to \mathbb{R}^m$ \\
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\vspace{2mm}
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In other words, \textbf{matrices are linear transformations}. \\
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If you only learn only one thing today, this should be it.
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\vfill
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\problem{}<prooffwd>
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Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\
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Before you start, answer the following questions:
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\begin{itemize}
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\item What is $A$?
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\item What is $v$?
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\item What are their sizes?
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\end{itemize}
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\vfill
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\problem{}<proofback>
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Show that any linear transformation can be written as a matrix.
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\vfill
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\pagebreak
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\problem{}
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Does \ref{thebigtheorem} hold in arbitrary vector spaces? \\
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Repeat \ref{prooffwd} and \ref{proofback} using only axioms.
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\vfill
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\pagebreak
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