Rearranged handout

Advanced/Linear Maps/main.tex

@@ -26,9 +26,6 @@
 			Prepared by Mark on \today \\
 		}
 
-	\section{Fields and Vector Spaces}
-
-
 	\input{parts/0 fields}
 	\input{parts/1 spaces}
 	\input{parts/2 linearity}
@@ -40,7 +37,7 @@
 	\section{Bonus}
 
 	\definition{}
-	Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space.
+	Show that $\mathbb{P}^n$ is a vector space.
 	\vfill
 
 	\problem{}

Advanced/Linear Maps/parts/0 fields.tex

@@ -1,3 +1,5 @@
+\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 $ab$. \\

Advanced/Linear Maps/parts/1 spaces.tex

@@ -1,3 +1,5 @@
+\section{Spaces}
+
 \definition{Vector Spaces}
 A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
 \begin{itemize}[itemsep = 2mm]
@@ -5,7 +7,7 @@ A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
 	\item An operation called \textit{vector addition}, denoted $+$ \\
 		Vector addition operates on two elements of $V$. \\
 
-	\item An operation called \textit{scalar multilplication}, denoted $\times$ \\
+	\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}

Advanced/Linear Maps/parts/2 linear.tex

@@ -1,4 +1,4 @@
-\section{Linearity}
+\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 $f: A \to B$.
@@ -36,7 +36,8 @@ 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 $\mathbb{P}^n$?
+Is $\frac{d}{dx}(p): \mathbb{P}^n \to \mathbb{P}^{n-1}$ a linear map on $\mathbb{P}^n$? \\
+\hint{$\mathbb{P}^n$ is the set of all polynomials of degree $n$.}
 
 \vfill
 \pagebreak

Advanced/Linear Maps/parts/3 matrices.tex

@@ -17,8 +17,8 @@ Draw a $3 \times 2$ matrix.
 \vfill
 
 \definition{}
-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.}
+We can define the product of a matrix $A$ and a vector $v$:
+
 $$
 Av =
 \begin{bmatrix}
@@ -34,7 +34,7 @@ Av =
 	4a + 5b + 6c
 \end{bmatrix}
 $$
-Look closely. Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
+Each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
 
 $$
 Av =
@@ -61,9 +61,9 @@ Compute the following:
 
 $$
 \begin{bmatrix}
-	2 & 9 \\
-	7 & 5 \\
-	3 & 4
+	1 & 2 \\
+	3 & 4 \\
+	5 & 6
 \end{bmatrix}
 \begin{bmatrix}
 	5 \\ 3
@@ -84,16 +84,16 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
 
 		$$
 		\begin{bmatrix}
-			2 & 9 \\
-			7 & 5 \\
-			3 & 4
+			1 & 2 \\
+			3 & 4 \\
+			5 & 6
 		\end{bmatrix}
 		\begin{bmatrix}
 			5 \\ 3
 		\end{bmatrix}
 		=
 		\begin{bmatrix}
-			37 \\ 50 \\ 27
+			11 \\ 27 \\ 43
 		\end{bmatrix}
 		$$
 	\end{center}
@@ -110,9 +110,9 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
 			left delimiter={[},
 			right delimiter={]}
 		] (A) {
-			2 & 9 \\
-			7 & 5 \\
+			1 & 2 \\
 			3 & 4 \\
+			5 & 6 \\
 		};
 
 		\node[
@@ -133,21 +133,21 @@ It is a bit more interesting to think of matrix-vector multiplication in the fol
 		\node[
 			fit=(A-1-2)(A-1-2),
 			inner xsep=8mm,inner ysep=0mm,
-			label=right:{$10 + 27 = 37$}
+			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:{$35 + 15 = 50$}
+			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:{$15 + 12 = 27$}
+			label=right:{$25 + 18 = 43$}
 		](N) {};
 		\draw[->, gray] ([xshift=3mm]A-3-2.east) -- (N);
 	\end{tikzpicture}
@@ -187,11 +187,6 @@ Before you start, answer the following questions:
 Show that any linear transformation can be written as a matrix.
 
 \vfill
-\pagebreak
-
-\problem{}
-Show that $\mathbb{P}^n$, the set of polynomials of degree $n$, is a vector space.
-\vfill
 
 
 \problem{}