diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex new file mode 100755 index 0000000..c8b742b --- /dev/null +++ b/Advanced/Linear Maps/main.tex @@ -0,0 +1,50 @@ +% use [nosolutions] flag to hide solutions. +% use [solutions] flag to show solutions. +\documentclass[ + solutions, + singlenumbering +]{../../resources/ormc_handout} + +\usepackage{tikz} +\usetikzlibrary{ + matrix, + decorations.pathreplacing, + calc, + positioning, + fit +} + +%\usepackage{lua-visual-debug} +\renewcommand{\arraystretch}{1.2} +\begin{document} + + \maketitle + + + {Linear Maps} + { + Prepared by Mark on \today \\ + } + + \section{Fields and Vector Spaces} + + + \input{parts/0 fields} + \input{parts/1 spaces} + \input{parts/2 linearity} + \input{parts/3 matrices} + + + + + \section{Bonus} + + \problem{} + Is the set of all linear maps a vector space? + \vfill + + \definition{} + Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space. + \vfill + +\end{document} \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/0 fields.tex b/Advanced/Linear Maps/parts/0 fields.tex new file mode 100644 index 0000000..4d42573 --- /dev/null +++ b/Advanced/Linear Maps/parts/0 fields.tex @@ -0,0 +1,47 @@ +\definition{Fields and Field Axioms} +A \textit{field} $\mathbb{F}$ consists of a set $A$ and two operations $+$ and $\times$. \\ +As usual, we may abbreviate $a \times b$ as $ab$. \\ +The following axioms must be satisfied for any $a, b, c \in \mathbb{F}$: + +\vspace{1mm} +\begin{center} +% @{} supresses the space between columns. +% @{=} makes = a column seperator. +\begin{tabular}{l | r@{=}l | r@{=}l} + \hline + \multicolumn{1}{|c|}{Name} & + \multicolumn{2}{c}{$+$} & + \multicolumn{2}{|c|}{$\times$} \\ + \hline + Closure & \multicolumn{2}{c|}{$a+b \in \mathbb{F}$} & \multicolumn{2}{c}{$ab \in \mathbb{F}$} \\ + Associativity & $(a+b)+c~$&$~a+b+c$ & $(ab)c~$&$~a(bc)$ \\ + Commutativity & $a+b~$&$~b+a$ & $ab~$&$~ba$ \\ + Distributivity & $a(b+c)~$&$~ab + ac$ & \multicolumn{2}{}{} \\ + Identity & $a+0~$&$~a$ & $1 \times a~$&$~a$ \\ + Inverses & $a + (-a)~$&$~0$ & $a \times a^{-1}~$&$~1$ +\end{tabular} +\end{center} + + +\problem{} +Show that all fields are groups. \\ +Convince yourself that not all groups are fields. + +\vfill + + +\problem{} +Is $\mathbb{Z}$ a field under our usual definitions of $+$ and $\times$? \\ +Which axioms does it satisfy, and which does it violate? + +\vfill + +\problem{} +Verify that $\mathbb{R}$ is a field. +\vfill + +\generic{Remark:} +We won't worry too much about fields this week. They simply provide a foundation for \textit{spaces}. \\ +As such, you may assume that we are working in $\mathbb{R}$ for the rest of this handout. + +\pagebreak \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/1 spaces.tex b/Advanced/Linear Maps/parts/1 spaces.tex new file mode 100644 index 0000000..c989c51 --- /dev/null +++ b/Advanced/Linear Maps/parts/1 spaces.tex @@ -0,0 +1,92 @@ +\definition{Vector Spaces} +A \textit{space} over a field $\mathbb{F}$ consists of the following elements: +\begin{itemize}[itemsep = 2mm] + \item A set $V$, the elements of which are called \textit{vectors} + \item An operation called \textit{vector addition}, denoted $+$ \\ + Vector addition operates on two elements of $V$. \\ + + \item An operation called \textit{scalar multilplication}, denoted $\times$ \\ + Scalar multiplication multiplies an element of $V$ by an element of $\mathbb{F}$. \\ + Any element of $\mathbb{F}$ is called a \textit{scalar}. +\end{itemize} + +\vspace{2mm} + +\textbf{Note:} +The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\ +Be careful, since \textit{these are different operations!} \\ +Make sure you're aware of the context of each $+$ and $\times$ as you work through this handout. + +\vspace{5mm} + +Vector addition and multiplication must have the following properties. \\ +Note that $x, y, z \in V$ and $a, b\in \mathbb{F}$. + +\vspace{2mm} + +% [t] and \vspace{0pt} ensure alignment at top +\begin{minipage}[t]{0.48\textwidth}\vspace{0pt} + \begin{center} + \begin{tabular}{l | r@{=}l } + \hline + \multicolumn{3}{|c|}{Properties of vector addition} \\ + \hline + Closure & \multicolumn{2}{c}{$x+y \in V$} \\ + Associativity & $(x+y)+z~$&$~x+y+z$ \\ + Commutativity & $x+y~$&$~y+x$ \\ + Distributivity & $x(y+z)~$&$~xy + xz$ \\ + Identity & $x+0~$&$~x$ \\ + Inverse & $x + (-x)~$&$~0$ + \end{tabular} + \end{center} +\end{minipage}% +\hfill% +\begin{minipage}[t]{0.48\textwidth}\vspace{0pt} + \begin{center} + \begin{tabular}{l | r@{=}l } + \hline + \multicolumn{3}{|c|}{Properties of vector multiplication} \\ + \hline + Closure & \multicolumn{2}{c}{$ax \in V$} \\ + Distributivity & $a(x+y)~$&$~ax+ay$ \\ + & $(a+b)x~$&$~ax+bx$ \\ + Compatibility$^*$ & $(ab)x~$&$~x(ba)$ \\ + Identity & $a+0~$&$~a$ + \end{tabular} +\end{center} +\end{minipage} + +\vspace{5mm} + +\definition{} +There is a good chance you are familiar with basic vector arithmetic. \\ +Here's a quick review: +\begin{itemize} + \item Scalar multiplication is done elementwise: $3 \times [a, b, c] = [3a, 3b, 3c]$. + \item Vector addition is similar: $[a, b, c] + [1, 2, 3] = [a+1,~b+2,~c+3]$. + \item Vector addition is not valid for vectors of different sizes. +\end{itemize} + + +\definition{} +We usually use the \textit{dot product} as our vector product. It is defined as follows. \\ +Given two vectors $a, b \in \mathbb{R}^n$, the dot product $a \cdot b$ is $\sum_1^n a_ib_i$. + +\vspace{2mm} + +In other words, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$, +$$ + a \cdot b = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32 +$$ +As you may expect, the dot product $ab$ is valid iff $a$ and $b$ are the same size. + + + +\problem{} +Show that the dot product satisfies the properties of a vector product listed above. \\ +Conclude that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$. + +\vfill +\pagebreak + + diff --git a/Advanced/Linear Maps/parts/2 linearity.tex b/Advanced/Linear Maps/parts/2 linearity.tex new file mode 100644 index 0000000..9382198 --- /dev/null +++ b/Advanced/Linear Maps/parts/2 linearity.tex @@ -0,0 +1,42 @@ +\section{Linearity} + +\definition{} +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 $f: A \to B$. + + +\definition{This one is worth remembering} +Let $V$ and $U$ be vector spaces, and let $f: V \to U$ be a map from $V$ to $U$. \\ +We say $f$ is \textit{linear} if it satisfies the following for any $v \in V$, $u \in U$, $a \in \mathbb{R}$: +\begin{itemize} + \item $f(u + v) = f(u) + f(v)$ + \item $f(au) = af(u)$ +\end{itemize} +In other words, $f$ is linear if it is \say{closed} under addition and scalar multiplication. + + +\problem{} +It is often convenient to combine the two conditions above into one. \\ +Show that $f(au + v) = af(u) + f(v)$ iff $f$ is linear. + +\vfill + +\problem{} +Is $f(x) = mx + b$ a linear map on $\mathbb{R}$? + +\vfill +\problem{} +In general, what does a linear map in $\mathbb{R}^n$ look like? + +\vfill + +\problem{} +Is $\text{median}(v): \mathbb{R}^n \to \mathbb{R}$ a linear map on $\mathbb{R}^n$? + + +\vfill + +\problem{} +Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? + +\vfill +\pagebreak \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/3 matrices.tex b/Advanced/Linear Maps/parts/3 matrices.tex new file mode 100644 index 0000000..104ce56 --- /dev/null +++ b/Advanced/Linear Maps/parts/3 matrices.tex @@ -0,0 +1,198 @@ +\section{Matrices} + +\definition{} +A \textit{matrix} is a two-dimensional array of numbers: \\ +$$ +A = +\begin{bmatrix} + 1 & 2 & 3 \\ + 4 & 5 & 6 +\end{bmatrix} +$$ +The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix. + +\problem{} +Draw a $3 \times 2$ matrix. + +\vfill + +\definition{Matrices as Transformations} +We can define the \say{product\footnotemark{}} of a matrix $A$ and a vector $v$: +\footnotetext{This is an uncommon word to use in this context. You will soon see why.} +$$ +Av = +\begin{bmatrix} + 1 & 2 & 3 \\ + 4 & 5 & 6 +\end{bmatrix} +\begin{bmatrix} + a \\ b \\ c +\end{bmatrix} += +\begin{bmatrix} + 1a + 2b + 3c \\ + 4a + 5b + 6c +\end{bmatrix} +$$ +Look closely. Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$: + +$$ +Av = +\begin{bmatrix} + \text{---} a_1 \text{---} \\ + \text{---} a_2 \text{---} +\end{bmatrix} +\begin{bmatrix} + | \\ + v \\ + | \\ +\end{bmatrix} += +\begin{bmatrix} + r_1v \\ + r_2v +\end{bmatrix} +$$ + +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. + +\problem{} +Compute the following \say{product}: + +$$ +\begin{bmatrix} + 2 & 9 \\ + 7 & 5 \\ + 3 & 4 +\end{bmatrix} +\begin{bmatrix} + 5 \\ 3 +\end{bmatrix} +$$ + +\vfill +\pagebreak + + +\generic{Remark:} +It is a bit more interesting to think of matrix-vector multiplication in the following way: \\ + +\begin{minipage}[t]{0.48\textwidth}\vspace{0pt} + \begin{center} + The problem: + \vspace{2mm} + + $$ + \begin{bmatrix} + 2 & 9 \\ + 7 & 5 \\ + 3 & 4 + \end{bmatrix} + \begin{bmatrix} + 5 \\ 3 + \end{bmatrix} + = + \begin{bmatrix} + 37 \\ 50 \\ 27 + \end{bmatrix} + $$ + \end{center} +\end{minipage}% +\hfill +\begin{minipage}[t]{0.48\textwidth}\vspace{0pt} + \begin{center} + Top-input, right-output: + \vspace{2mm} + + \begin{tikzpicture}[>=stealth,thick,baseline] + \matrix [ + matrix of math nodes, + left delimiter={[}, + right delimiter={]} + ] (A) { + 2 & 9 \\ + 7 & 5 \\ + 3 & 4 \\ + }; + + \node[ + fit=(A-1-1)(A-1-1), + inner xsep=0mm,inner ysep=3mm, + label=above:5 + ] (L) {}; + \draw[->, gray] (L.north) -- ([yshift=0mm]A-1-1.north); + + \node[ + fit=(A-1-2)(A-1-2), + inner xsep=0mm,inner ysep=3mm, + label=above:3 + ] (R) {}; + \draw[->, gray] (R.north) -- ([yshift=0mm]A-1-2.north); + + + \node[ + fit=(A-1-2)(A-1-2), + inner xsep=8mm,inner ysep=0mm, + label=right:{$10 + 27 = 37$} + ](Y) {}; + \draw[->, gray] ([xshift=3mm]A-1-2.east) -- (Y); + + \node[ + fit=(A-2-2)(A-2-2), + inner xsep=8mm,inner ysep=0mm, + label=right:{$35 + 15 = 50$} + ](H) {}; + \draw[->, gray] ([xshift=3mm]A-2-2.east) -- (H); + + \node[ + fit=(A-3-2)(A-3-2), + inner xsep=8mm,inner ysep=0mm, + label=right:{$15 + 12 = 27$} + ](N) {}; + \draw[->, gray] ([xshift=3mm]A-3-2.east) -- (N); + \end{tikzpicture} + \end{center} +\end{minipage}% + +\vspace{2mm} + +Be aware that this is only a model for intuition. \\ +Make sure you understand the dot product definition on the previous page. + +\vspace{5mm} + +\theorem{} +Any linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as an $n \times m$ matrix. \\ +Conversely, every $n \times m$ matrix represents a liner map $T: \mathbb{R}^n \to \mathbb{R}^m$ \\ + +\vspace{2mm} + +In other words, \textbf{matrices are linear transformations}. \\ +If you only learn only one thing today, this should be it. + +\vfill + +\problem{} +Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\ +Before you start, answer the following questions: +\begin{itemize} + \item What is $A$? + \item What is $v$? + \item What are their sizes? +\end{itemize} + +\vfill + +\problem{} +Show that any linear transformation can be written as a matrix. + +\vfill +\pagebreak + + +\problem{} +Does \ref{thebigtheorem} hold in arbitrary vector spaces? \\ +Repeat \ref{prooffwd} and \ref{proofback} using only axioms. + +\vfill +\pagebreak \ No newline at end of file