From 2e3e1b3c5600a32b08ae562f1c90d5b83414641a Mon Sep 17 00:00:00 2001 From: Mark Date: Wed, 3 May 2023 22:40:03 -0700 Subject: [PATCH] Removed linear map handout --- Advanced/Linear Maps/main.tex | 49 ---------- Advanced/Linear Maps/parts/0 intro.tex | 6 -- Advanced/Linear Maps/parts/1 linear.tex | 58 ----------- Advanced/Linear Maps/parts/2 matrices.tex | 34 ------- Advanced/Linear Maps/parts/?0 fields.tex | 49 ---------- Advanced/Linear Maps/parts/?1 spaces.tex | 113 ---------------------- 6 files changed, 309 deletions(-) delete mode 100755 Advanced/Linear Maps/main.tex delete mode 100644 Advanced/Linear Maps/parts/0 intro.tex delete mode 100644 Advanced/Linear Maps/parts/1 linear.tex delete mode 100644 Advanced/Linear Maps/parts/2 matrices.tex delete mode 100644 Advanced/Linear Maps/parts/?0 fields.tex delete mode 100644 Advanced/Linear Maps/parts/?1 spaces.tex diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex deleted file mode 100755 index 26d11a2..0000000 --- a/Advanced/Linear Maps/main.tex +++ /dev/null @@ -1,49 +0,0 @@ -% use [nosolutions] flag to hide solutions. -% use [solutions] flag to show solutions. -\documentclass[ - solutions, - nowarning, - singlenumbering -]{../../resources/ormc_handout} - -\usepackage{tikz} -\usetikzlibrary{ - matrix, - decorations.pathreplacing, - calc, - positioning, - fit -} - - -% Let's give clarifications about the meaning of Z and R when we use them in the first problems. - -% Definitely define $R^n$ before using. Optionally you may add a problem "convince yourself that $R^2$ is a plane and $R^3$ is a 3-d space". - -% Maybe we can add an example of a linear transformation from R^2 to R^2? Rotation? Scaling of y-axis? - -% Slow down, understand linear transformations fully. - - - - -%\usepackage{lua-visual-debug} -\renewcommand{\arraystretch}{1.2} -\begin{document} - - \maketitle - - - {Linear Maps} - { - Prepared by Mark on \today \\ - } - - %\input{parts/? fields} - %\input{parts/? spaces} - - \input{parts/0 intro} - \input{parts/1 linear} - \input{parts/2 matrices} - -\end{document} \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/0 intro.tex b/Advanced/Linear Maps/parts/0 intro.tex deleted file mode 100644 index bffaa69..0000000 --- a/Advanced/Linear Maps/parts/0 intro.tex +++ /dev/null @@ -1,6 +0,0 @@ -\section{Intro} - - - -\vfill -\pagebreak \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/1 linear.tex b/Advanced/Linear Maps/parts/1 linear.tex deleted file mode 100644 index 74c4809..0000000 --- a/Advanced/Linear Maps/parts/1 linear.tex +++ /dev/null @@ -1,58 +0,0 @@ -\section{Linear Maps} - -\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{} -Let $f: U \to V$ be a map. \\ -We say $f$ is \textit{linear} if it satisfies the following for any $u \in U$, $v \in V$, $a \in \mathbb{R}$: -\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. Use \ref{lineardef}. - -\vfill - -\problem{} -Is $f(x) = mx + b$ a linear map? - -\vfill - -\problem{} -In general, what does a linear map in $\mathbb{R} \to \mathbb{R}$ look like? - -\vfill -\pagebreak - -\problem{} -Is the map ${median}(v): \mathbb{R}^3 \to \mathbb{R}$ linear? \\ -\hint{$median([3, 5, 4]) = 4$, but you already knew that.} - -\vfill - -\problem{} -Is the map $f(v): \mathbb{R}^3 \to \mathbb{R}$ defined by $f(v) = v_0 + 2v_1 + v_2$ linear? \\ -\hint{$v_n$ is the $n^\text{th}$ element of $v$. $v$ is a 3-element vector.} - - -\vfill - -\problem{} -Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\ -\vspace{1mm} -\hint{$\mathbb{P}^n$ is the set of polynomials with degree at most $n$.} - -\vfill - -\problem{} -In general, what does a linear map from $\mathbb{R}^m \to \mathbb{R}^n$ look like? - -\vfill -\pagebreak \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/2 matrices.tex b/Advanced/Linear Maps/parts/2 matrices.tex deleted file mode 100644 index 59ab943..0000000 --- a/Advanced/Linear Maps/parts/2 matrices.tex +++ /dev/null @@ -1,34 +0,0 @@ -\section{Matrices} - -\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 linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ \\ - -\vspace{2mm} - -In other words, \textbf{matrices are linear transformations}. \\ -The next two problems provide a proof. - - -\problem{} -Let $A$ be an $m \times n$ matrix, and $v$ an $m \times 1$ vector. \\ -Show that the map $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\ - - - -\vfill - -\problem{} -Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$. - - -\vfill -\pagebreak - -\problem{} -Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}(p)$. \\ -Find a matrix that corresponds to $D$. \\ -\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?} - -\vfill -\pagebreak \ No newline at end of file diff --git a/Advanced/Linear Maps/parts/?0 fields.tex b/Advanced/Linear Maps/parts/?0 fields.tex deleted file mode 100644 index 0df8b72..0000000 --- a/Advanced/Linear Maps/parts/?0 fields.tex +++ /dev/null @@ -1,49 +0,0 @@ -\section{Fields} - -\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 deleted file mode 100644 index a636410..0000000 --- a/Advanced/Linear Maps/parts/?1 spaces.tex +++ /dev/null @@ -1,113 +0,0 @@ -\section{Spaces} - -\definition{} -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 multiplication}, 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 $V$. \\ -\textit{These are different operations}, so be aware of the context of each $+$ and $\times$. - -\vspace{5mm} - -Vector addition and multiplication must have the following properties. \\ -In both tables, $x, y, z \in V$ 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 scalar multiplication} \\ - \hline - Closure & \multicolumn{2}{c}{$ax \in V$} \\ - Distributivity & $a(x+y)~$&$~ax+ay$ \\ - & $(a+b)x~$&$~ax+bx$ \\ - Compatibility$^*$ & $(ab)x~$&$~a(bx)$ \\ - Identity & $a+0~$&$~a$ - \end{tabular} -\end{center} -\end{minipage} - -\vspace{2mm} - -$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is multiplication in $\mathbb{F}$ and $(bx)$ is scalar multiplication in $V$. Compatibility is \textit{not} associativity. \\ - -Likewise, the addition you see in the distributive property of multiplication is field addition, not vector addition. - -\vspace{6mm} - -Usually, the word \textit{vector} refers to an element of $\mathbb{R}^n$. As you might expect $\mathbb{R}^n$ is a vector space over the field $\mathbb{R}$ under our usual vector operations. - -Here's a quick review of these operations: -\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} - -\problem{} -Verify that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$ under these operations. - -\vfill - -\pagebreak - -We can also define an \textit{inner product} or \textit{vector product} that takes two elements of $V$ and produces another. \\ - -When we work in $\mathbb{R}^n$, we usually use the dot product as our vector product. It is defined as follows: \\ - -\definition{Dot Product} -Given two vectors $a, b \in \mathbb{R}^n$, the \textit{dot product} of $a$ and $b$ (written $a \cdot b$ or $\langle a, b \rangle$) is $\sum_1^n a_ib_i$. - -\vspace{2mm} - -For example, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$, -$$ - \langle a, b \rangle = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32 -$$ -As you may expect, the dot product $\langle a, b \rangle$ is valid iff $a$ and $b$ are the same size. - - - -\problem{} -Show that the dot product is commutative. - -\vfill - -\problem{} -Show that the dot product is positive-definite. \\ -This means that $\langle a, a \rangle > 0$ unless $a = 0$. - -\vfill -\pagebreak - -