From 5b08e1d22427179cd4f0f0591365ca72c8e70ec1 Mon Sep 17 00:00:00 2001 From: Mark Date: Mon, 17 Apr 2023 11:34:36 -0700 Subject: [PATCH] Rearranged Linear Maps --- Advanced/Linear Maps/main.tex | 51 +---- Advanced/Linear Maps/parts/0 intro.tex | 6 + Advanced/Linear Maps/parts/1 linear.tex | 58 +++++ Advanced/Linear Maps/parts/2 linear.tex | 38 ---- Advanced/Linear Maps/parts/2 matrices.tex | 34 +++ Advanced/Linear Maps/parts/3 matrices.tex | 209 ------------------ .../parts/{0 fields.tex => ?0 fields.tex} | 0 .../parts/{1 spaces.tex => ?1 spaces.tex} | 0 8 files changed, 104 insertions(+), 292 deletions(-) create mode 100644 Advanced/Linear Maps/parts/0 intro.tex create mode 100644 Advanced/Linear Maps/parts/1 linear.tex delete mode 100644 Advanced/Linear Maps/parts/2 linear.tex create mode 100644 Advanced/Linear Maps/parts/2 matrices.tex delete mode 100644 Advanced/Linear Maps/parts/3 matrices.tex rename Advanced/Linear Maps/parts/{0 fields.tex => ?0 fields.tex} (100%) rename Advanced/Linear Maps/parts/{1 spaces.tex => ?1 spaces.tex} (100%) diff --git a/Advanced/Linear Maps/main.tex b/Advanced/Linear Maps/main.tex index be3e488..26d11a2 100755 --- a/Advanced/Linear Maps/main.tex +++ b/Advanced/Linear Maps/main.tex @@ -3,7 +3,7 @@ \documentclass[ solutions, nowarning, - %singlenumbering + singlenumbering ]{../../resources/ormc_handout} \usepackage{tikz} @@ -39,50 +39,11 @@ Prepared by Mark on \today \\ } - \input{parts/0 fields} - \input{parts/1 spaces} - \input{parts/2 linear} - \input{parts/3 matrices} + %\input{parts/? fields} + %\input{parts/? spaces} - - \section{Norms} - - \definition{} - If $V$ is a vector space, a \textit{norm} in $V$ is a function $V \to \mathbb{R}^+$ that satisfies the following properties, \\ - Where $x, y \in V$ and $c \in F$: - - \begin{itemize} - \item Absolute Homogeneity: $||cx|| = |c|~||x||$ - \item Positive-Definite: $||x|| \geq 0$ with equality iff $x = 0$. - \item Triangle Inequalty: $||x+y|| \leq ||x|| + ||y||$ - \end{itemize} - - \problem{} - Show that the \textit{euclidian norm} defined by $||~[a, b]~|| = \sqrt{a^2 + b^2}$ is a norm on $\mathbb{R}^2$ - - \vfill - - \problem{} - Show that in any vector space with an inner product, the \textit{induced norm} $||x|| = \sqrt{\langle x, x \rangle}$ is a norm. - - \vfill - - \problem{} - Show that every norm satisfies the reverse triangle inequality: - - $$ - ||x - y|| \geq |~||x|| - ||y||~| - $$ - - \vfill - - \problem{} - Prove the Cauchy-Schwartz inequality: - - $$ - ||\langle x, y \rangle|| = ||x||~||y|| - $$ - - \vfill + \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 new file mode 100644 index 0000000..bffaa69 --- /dev/null +++ b/Advanced/Linear Maps/parts/0 intro.tex @@ -0,0 +1,6 @@ +\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 new file mode 100644 index 0000000..74c4809 --- /dev/null +++ b/Advanced/Linear Maps/parts/1 linear.tex @@ -0,0 +1,58 @@ +\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 linear.tex b/Advanced/Linear Maps/parts/2 linear.tex deleted file mode 100644 index 8c2be8d..0000000 --- a/Advanced/Linear Maps/parts/2 linear.tex +++ /dev/null @@ -1,38 +0,0 @@ -\section{Linear Transformations} - -\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 $V$ 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 $\mathbb{R} \to \mathbb{R}$ look like? - -\vfill - -\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 -\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 new file mode 100644 index 0000000..59ab943 --- /dev/null +++ b/Advanced/Linear Maps/parts/2 matrices.tex @@ -0,0 +1,34 @@ +\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/3 matrices.tex b/Advanced/Linear Maps/parts/3 matrices.tex deleted file mode 100644 index d3b2bea..0000000 --- a/Advanced/Linear Maps/parts/3 matrices.tex +++ /dev/null @@ -1,209 +0,0 @@ -\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. - -\definition{} -We can define the product of a matrix $A$ and a vector $v$: - -$$ -Av = -\begin{bmatrix} - 1 & 2 & 3 \\ - 4 & 5 & 6 -\end{bmatrix} -\begin{bmatrix} - a \\ b \\ c -\end{bmatrix} -= -\begin{bmatrix} - 1a + 2b + 3c \\ - 4a + 5b + 6c -\end{bmatrix} -$$ -Note that 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{} -Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result? - -\vfill - - -\problem{} -Compute the following: - -$$ -\begin{bmatrix} - 1 & 2 \\ - 3 & 4 \\ - 5 & 6 -\end{bmatrix} -\begin{bmatrix} - 5 \\ 3 -\end{bmatrix} -$$ - -\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} - - $$ - \begin{bmatrix} - 1 & 2 \\ - 3 & 4 \\ - 5 & 6 - \end{bmatrix} - \begin{bmatrix} - 5 \\ 3 - \end{bmatrix} - = - \begin{bmatrix} - 11 \\ 27 \\ 43 - \end{bmatrix} - $$ - \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) { - 1 & 2 \\ - 3 & 4 \\ - 5 & 6 \\ - }; - - \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:{$5 + 6 = 11$} - ](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:{$15 + 12 = 27$} - ](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:{$25 + 18 = 43$} - ](N) {}; - \draw[->, gray] ([xshift=3mm]A-3-2.east) -- (N); - \end{tikzpicture} - \end{center} -\end{minipage}% - -\vspace{2mm} - -This is only a model for intuition, though. \\ -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}. -\problem{} -Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\ -\hint{What is $A$? What is $v$? What are their sizes?} - -\vfill - -\problem{} -Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$. - -\vfill - -\vfill -\pagebreak - -\problem{} -Show that $\mathbb{P}^n$ is a vector space. -\hint{$\mathbb{P}^n$ is the set of all polynomials of degree $ \leq n$.} - -\vfill - -\problem{} -Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\ - -\vfill - -\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 - -\problem{} -Show that the set of all linear maps $\mathbb{R}^n \to \mathbb{R}^m$ is a vector space. - -\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 similarity index 100% rename from Advanced/Linear Maps/parts/0 fields.tex rename to Advanced/Linear Maps/parts/?0 fields.tex diff --git a/Advanced/Linear Maps/parts/1 spaces.tex b/Advanced/Linear Maps/parts/?1 spaces.tex similarity index 100% rename from Advanced/Linear Maps/parts/1 spaces.tex rename to Advanced/Linear Maps/parts/?1 spaces.tex