284 lines
5.0 KiB
TeX
284 lines
5.0 KiB
TeX
\section{Matrices}
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\definition{}
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A \textit{matrix} is a two-dimensional array of numbers: \\
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$$
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A =
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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$$
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The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix. \\
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\vspace{1mm}
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The order \say{first rows, then columns} is usually consistent in linear algebra. \\
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If you look closely, you may also find it in the next definition.
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\definition{}<matvec>
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We can define the product of a matrix $A$ and a vector $v$:
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$$
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Av =
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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a \\ b \\ c
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\end{bmatrix}
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=
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\begin{bmatrix}
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1a + 2b + 3c \\
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4a + 5b + 6c
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\end{bmatrix}
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$$
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Note that each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
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$$
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Av =
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\begin{bmatrix}
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\text{---} r_1 \text{---} \\
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\text{---} r_2 \text{---}
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\end{bmatrix}
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\begin{bmatrix}
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| \\
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v \\
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| \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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r_1 \cdot v \\
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r_2 \cdot v
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\end{bmatrix}
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$$
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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.
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\problem{}
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Compute the following:
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$$
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\begin{bmatrix}
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1 & 2 \\
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3 & 4 \\
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5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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5 \\ 3
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\end{bmatrix}
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$$
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\vfill
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\problem{}
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Say you multiply a size-$m$ vector $v$ by an $m \times n$ matrix $A$. \\
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What is the size of your result $Av$?
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\vfill
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\pagebreak
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\definition{}
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We can also multiply a matrix by a matrix:
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$$
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AB =
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\begin{bmatrix}
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1 & 2 \\
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3 & 4
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\end{bmatrix}
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\begin{bmatrix}
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10 & 20 \\
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100 & 200
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\end{bmatrix}
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=
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\begin{bmatrix}
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210 & 420 \\
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430 & 860
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\end{bmatrix}
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$$
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Note each element of the resulting matrix is dot product of a row of $A$ and a column of $B$:
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$$
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AB =
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\begin{bmatrix}
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\text{---} r_1 \text{---} \\
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\text{---} r_2 \text{---}
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\end{bmatrix}
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\begin{bmatrix}
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| & | \\
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v_1 & v_2 \\
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| & | \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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r_1 \cdot v_1 & r_1 \cdot v_2 \\
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r_2 \cdot v_1 & r_2 \cdot v_2 \\
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\end{bmatrix}
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$$
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\begin{center}
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\begin{tikzpicture}
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\begin{scope}[layer = nodes]
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\matrix[
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matrix of math nodes,
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left delimiter={[},
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right delimiter={]}
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] (A) at (0, 0){
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1 & 2 \\
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3 & 4 \\
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};
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\matrix[
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matrix of math nodes,
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left delimiter={[},
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right delimiter={]}
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] (B) at (2, 0) {
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10 & 20 \\
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100 & 200 \\
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};
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\node at (3.25, 0) {$=$};
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\matrix[
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matrix of math nodes,
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left delimiter={[},
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right delimiter={]}
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] (C) at (4.5, 0) {
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210 & 420 \\
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430 & 860 \\
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};
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\end{scope}
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\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-2mm,yshift=2mm]A-1-1) rectangle ([xshift=2mm,yshift=-2mm]A-1-2) {};
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\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-2-1) {};
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\draw[rounded corners,fill=black!30!white,draw=none] ([xshift=-4mm,yshift=2mm]C-1-1) rectangle ([xshift=4mm,yshift=-2mm]C-1-1) {};
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\draw[rounded corners] ([xshift=-2mm,yshift=2mm]A-2-1) rectangle ([xshift=2mm,yshift=-2mm]A-2-2) {};
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\draw[rounded corners] ([xshift=-3mm,yshift=2mm]B-1-2) rectangle ([xshift=3mm,yshift=-2mm]B-2-2) {};
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\draw[rounded corners] ([xshift=-4mm,yshift=2mm]C-2-2) rectangle ([xshift=4mm,yshift=-2mm]C-2-2) {};
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\end{tikzpicture}
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\end{center}
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\problem{}
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Compute the following matrix product. \\
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$$
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\begin{bmatrix}
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1 & 2 \\
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3 & 4 \\
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5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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9 & 8 & 7 \\
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6 & 5 & 4
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\end{bmatrix}
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$$
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\vfill
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\problem{}
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Compute the following matrix product or explain why you can't.
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$$
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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10 & 20 \\
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30 & 40
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\end{bmatrix}
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$$
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\vfill
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\problem{}
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If $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, when does the product $AB$ exist?
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\vfill
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\pagebreak
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\problem{}
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Is matrix multiplication commutative? \\
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\note{Does $AB = BA$ for all $A, B$? \\ You only need one counterexample to show this is false.}
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\vfill
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\definition{}
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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:
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$$
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\begin{bmatrix}
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1 & 2 & 3 \\
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4 & 5 & 6
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\end{bmatrix} ^ T
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=
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\begin{bmatrix}
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1 & 4 \\
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2 & 5 \\
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3 & 6
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\end{bmatrix}
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$$
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\problem{}
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Compute the following:
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\hfill
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$
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\begin{bmatrix}
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a & b \\
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c & d
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\end{bmatrix} ^ T
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$\hfill
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$
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\begin{bmatrix}
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1 \\
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3 \\
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3 \\
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7 \\
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\end{bmatrix} ^ T
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$\hfill
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$
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\begin{bmatrix}
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1 & 2 & 4 & 8 \\
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\end{bmatrix} ^ T
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$
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\hfill~
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\vfill
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\pagebreak
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The \say{transpose} operator is often used to write column vectors in a compact way. \\
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Vertical arrays don't look good in horizontal text.
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\problem{}
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Consider the vectors $a = [1, 4, 3]^T$ and $b = [9, 1, 4]^T$ \\
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\begin{itemize}
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\item Compute the dot product $a \cdot b$.
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\item Can you redefine the dot product using matrix multiplication?
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\end{itemize}
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\note{As you may have noticed, a vector is a special case of a matrix.}
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\vfill
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\problem{}
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A \textit{column vector} is an $m \times 1$ matrix. \\
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A \textit{row vector} is a $1 \times m$ matrix. \\
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We usually use column vectors. Why? \\
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\hint{How does vector-matrix multiplication work?}
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\vfill
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\pagebreak |