Rewrote Vectors, added Norms

Advanced/Linear Maps/main.tex

@@ -2,7 +2,8 @@
 % use [solutions] flag to show solutions.
 \documentclass[
 	solutions,
-	singlenumbering
+	nowarning,
+	%singlenumbering
 ]{../../resources/ormc_handout}
 
 \usepackage{tikz}
@@ -32,16 +33,44 @@
 	\input{parts/3 matrices}
 
 
-
+	\section{Norms}
-
-	\section{Bonus}
 
 	\definition{}
-	Show that $\mathbb{P}^n$ is a vector space.
+	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 the set of all linear maps is a vector space.
+	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/1 spaces.tex

@@ -1,6 +1,6 @@
 \section{Spaces}
 
-\definition{Vector 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}
@@ -46,7 +46,7 @@ In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$.
 	\begin{center}
 	\begin{tabular}{l | r@{=}l }
 		\hline
-		\multicolumn{3}{|c|}{Properties of vector multiplication} \\
+		\multicolumn{3}{|c|}{Properties of scalar multiplication} \\
 		\hline
 		Closure			& \multicolumn{2}{c}{$ax \in V$} \\
 		Distributivity	& $a(x+y)~$&$~ax+ay$	\\
@@ -59,39 +59,53 @@ In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$.
 
 \vspace{2mm}
 
-$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is scalar multiplication and $(bx)$ is vector multiplication. \\
+$^*$ 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. \\
-Compatibility is \textit{not} associativity, it is \say{compatibility of vector and scalar multiplication.}
 
-\vfill
+Likewise, the addition you see in the distributive property of multiplication is field addition, not vector addition.
-\pagebreak
 
-\definition{}
+\vspace{6mm}
-There is a good chance you are familiar with basic vector arithmetic. \\
+
-Here's a quick review:
+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.
 
-\definition{}
+\vfill
-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 $\sum_1^n a_ib_i$.
+\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}
 
-In other words, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
+For example, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
 $$
-	a \cdot b = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
+	\langle a, b \rangle = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
 $$
-As you may expect, the dot product $ab$ is valid iff $a$ and $b$ are the same size.
+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 satisfies the properties of a vector product listed above. \\
+Show that the dot product is commutative.
-Conclude that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$.
+
+\vfill
+
+\problem{}
+Show that the dot product is positive-definite. \\
+This means that $\langle a, a \rangle > 0$ unless $a = 0$.
 
 \vfill
 \pagebreak

Advanced/Linear Maps/parts/3 matrices.tex

@@ -29,7 +29,7 @@ Av =
 	4a + 5b + 6c
 \end{bmatrix}
 $$
-Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
+Note that each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
 
 $$
 Av =
@@ -191,10 +191,8 @@ Find a matrix that corresponds to $D$. \\
 \vfill
 \pagebreak
 
-
 \problem{}
-Does \ref{thebigtheorem} hold in arbitrary vector spaces? \\
+Show that the set of all linear maps is a vector space.
-Repeat \ref{prooffwd} and \ref{proofback} using only axioms, without assuming that we're working in $\mathbb{R}^n$.
 
 \vfill
 \pagebreak