From 81328c02e2d4b0488d977a3910b64877ab83423f Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Mon, 17 Apr 2023 09:26:08 -0700
Subject: [PATCH] Finished LA 101 handout

---
 .../Linear Algebra 101/parts/3 matrices.tex   | 84 ++++++++++++++++---
 1 file changed, 73 insertions(+), 11 deletions(-)

diff --git a/Advanced/Linear Algebra 101/parts/3 matrices.tex b/Advanced/Linear Algebra 101/parts/3 matrices.tex
index be611a3..1431de7 100644
--- a/Advanced/Linear Algebra 101/parts/3 matrices.tex	
+++ b/Advanced/Linear Algebra 101/parts/3 matrices.tex	
@@ -115,7 +115,7 @@ AB =
 $$
 
 \begin{center}
-\begin{tikzpicture}[>=stealth,thick,baseline]
+\begin{tikzpicture}
 
 	\begin{scope}[layer = nodes]
 		\matrix[
@@ -148,16 +148,16 @@ $$
 		};
 	\end{scope}
 
-	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-2mm,yshift=3mm]A-1-1) rectangle ([xshift=2mm,yshift=-3mm]A-2-1) {};
+	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-2mm,yshift=2mm]A-1-1) rectangle ([xshift=2mm,yshift=-2mm]A-1-2) {};
 
-	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-1-2) {};
+	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-2-1) {};
 
 	\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-4mm,yshift=2mm]C-1-1) rectangle ([xshift=4mm,yshift=-2mm]C-1-1) {};
 
 
-	\draw[rounded corners] ([xshift=-2mm,yshift=3mm]A-1-2) rectangle ([xshift=2mm,yshift=-3mm]A-2-2) {};
+	\draw[rounded corners] ([xshift=-2mm,yshift=2mm]A-2-1) rectangle ([xshift=2mm,yshift=-2mm]A-2-2) {};
 
-	\draw[rounded corners] ([xshift=-3mm,yshift=2mm]B-2-1) rectangle ([xshift=3mm,yshift=-2mm]B-2-2) {};
+	\draw[rounded corners] ([xshift=-3mm,yshift=2mm]B-1-2) rectangle ([xshift=3mm,yshift=-2mm]B-2-2) {};
 
 	\draw[rounded corners] ([xshift=-4mm,yshift=2mm]C-2-2) rectangle ([xshift=4mm,yshift=-2mm]C-2-2) {};
 \end{tikzpicture}
@@ -183,8 +183,7 @@ $$
 
 
 \problem{}
-Consider the following matrix product. \\
-Compute it or explain why you can't.
+Compute the following matrix product or explain why you can't.
 
 $$
 \begin{bmatrix}
@@ -207,13 +206,76 @@ If $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, when does the
 \vfill
 \pagebreak
 
+\problem{}
+Is matrix multiplication commutative? \\
+\note{Does $AB = BA$ for all $A, B$? \\ You only need one counterexample to show this is false.}
+
+\vfill
+
+
+\definition{}
+Say we have a matrix $A$. The matrix $A^T$, pronounced \say{A-transpose}, is created by turning rows of $A$ into columns, and columns into rows:
+
+$$
+\begin{bmatrix}
+	1 & 2 & 3 \\
+	4 & 5 & 6
+\end{bmatrix} ^ T
+=
+\begin{bmatrix}
+	1 & 4 \\
+	2 & 5 \\
+	3 & 6
+\end{bmatrix}
+$$
 
 \problem{}
-Look back to \ref{matvec}. \\
-Convince yourself that vectors are matrices. \\
+Compute the following:
+
+\hfill
+$
+\begin{bmatrix}
+	a & b \\
+	c & d
+\end{bmatrix} ^ T
+$\hfill
+$
+\begin{bmatrix}
+	1 \\
+	3 \\
+	3 \\
+	7 \\
+\end{bmatrix} ^ T
+$\hfill
+$
+\begin{bmatrix}
+	1 & 2 & 4 & 8 \\
+\end{bmatrix} ^ T
+$
+\hfill~
+
+\vfill
+\pagebreak
+
+The \say{transpose} operator is often used to write column vectors compactly. \\
+Vertical arrays don't look good in horizontal text.
+
+\problem{}
+Consider the vectors $a = [1, 2, 3]^T$ and $b = [40, 50, 60]^T$ \\
+\begin{itemize}
+	\item Compute the dot product $ab$.
+	\item Can you redefine the dot product using matrix multiplication?
+\end{itemize}
+\note{As you may have noticed, a vector is a special case of a matrix.}
+
+\vfill
+
+\problem{}
+A \textit{column vector} is an $m \times 1$ matrix. \\
+A \textit{row vector} is a $1 \times m$ matrix. \\
+We usually use column vectors. Why? \\
+\hint{How does vector-matrix multiplication work?}
 
-Can you multiply a matrix by a vector, as in $vA$? \\
-How does the dot prouduct relate to matrix multiplication? (transpose)
 
 \vfill
 \pagebreak
\ No newline at end of file