Rearranged Linear Maps

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/? fields}
-	\input{parts/1 spaces}
+	%\input{parts/? spaces}
-	\input{parts/2 linear}
-	\input{parts/3 matrices}
 
-
+	\input{parts/0 intro}
-	\section{Norms}
+	\input{parts/1 linear}
-
+	\input{parts/2 matrices}
-	\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
 
 \end{document}

Advanced/Linear Maps/parts/0 intro.tex

@@ -0,0 +1,6 @@
+\section{Intro}
+
+
+
+\vfill
+\pagebreak

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{}<lineardef>
+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

Advanced/Linear Maps/parts/2 linear.tex

@@ -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

Advanced/Linear Maps/parts/2 matrices.tex

@@ -0,0 +1,34 @@
+\section{Matrices}
+
+\theorem{}<thebigtheorem>
+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{}<prooffwd>
+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{}<proofback>
+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

Advanced/Linear Maps/parts/3 matrices.tex

@@ -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{}<thebigtheorem>
-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{}<prooffwd>
-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{}<proofback>
-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