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Finished LA 101 handout

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Mark 2023-04-17 09:26:08 -07:00
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84
Advanced/Linear Algebra 101/parts/3 matrices.tex
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@ -115,7 +115,7 @@ AB =
$$ $$
\begin{center} \begin{center}
\begin{tikzpicture}[>=stealth,thick,baseline] \begin{tikzpicture}
\begin{scope}[layer = nodes] \begin{scope}[layer = nodes]
\matrix[ \matrix[
@ -148,16 +148,16 @@ $$
}; };
\end{scope} \end{scope}
\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-2mm,yshift=3mm]A-1-1) rectangle ([xshift=2mm,yshift=-3mm]A-2-1) {}; \draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-2mm,yshift=2mm]A-1-1) rectangle ([xshift=2mm,yshift=-2mm]A-1-2) {};
\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-1-2) {}; \draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-2-1) {};
\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-4mm,yshift=2mm]C-1-1) rectangle ([xshift=4mm,yshift=-2mm]C-1-1) {}; \draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-4mm,yshift=2mm]C-1-1) rectangle ([xshift=4mm,yshift=-2mm]C-1-1) {};
\draw[rounded corners] ([xshift=-2mm,yshift=3mm]A-1-2) rectangle ([xshift=2mm,yshift=-3mm]A-2-2) {}; \draw[rounded corners] ([xshift=-2mm,yshift=2mm]A-2-1) rectangle ([xshift=2mm,yshift=-2mm]A-2-2) {};
\draw[rounded corners] ([xshift=-3mm,yshift=2mm]B-2-1) rectangle ([xshift=3mm,yshift=-2mm]B-2-2) {}; \draw[rounded corners] ([xshift=-3mm,yshift=2mm]B-1-2) rectangle ([xshift=3mm,yshift=-2mm]B-2-2) {};
\draw[rounded corners] ([xshift=-4mm,yshift=2mm]C-2-2) rectangle ([xshift=4mm,yshift=-2mm]C-2-2) {}; \draw[rounded corners] ([xshift=-4mm,yshift=2mm]C-2-2) rectangle ([xshift=4mm,yshift=-2mm]C-2-2) {};
\end{tikzpicture} \end{tikzpicture}
@ -183,8 +183,7 @@ $$
\problem{} \problem{}
Consider the following matrix product. \\ Compute the following matrix product or explain why you can't.
Compute it or explain why you can't.
$$ $$
\begin{bmatrix} \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 \vfill
\pagebreak \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{} \problem{}
Look back to \ref{matvec}. \\ Compute the following:
Convince yourself that vectors are matrices. \\
\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 \vfill
\pagebreak \pagebreak