198 lines
3.8 KiB
TeX
198 lines
3.8 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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\definition{}
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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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$$
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Av =
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\begin{bmatrix}
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\text{---} a_1 \text{---} \\
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\text{---} a_2 \text{---}
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\end{bmatrix}
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\begin{bmatrix}
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| \\
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v \\
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| \\
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\end{bmatrix}
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=
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\begin{bmatrix}
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r_1v \\
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r_2v
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\end{bmatrix}
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$$
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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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Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result?
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\vfill
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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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\pagebreak
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\generic{Remark:}
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It is a bit more interesting to think of matrix-vector multiplication in the following way: \\
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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The problem:
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\vspace{2mm}
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$$
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\begin{bmatrix}
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1 & 2 \\
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3 & 4 \\
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5 & 6
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\end{bmatrix}
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\begin{bmatrix}
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5 \\ 3
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\end{bmatrix}
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=
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\begin{bmatrix}
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11 \\ 27 \\ 43
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\end{bmatrix}
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$$
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\end{center}
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\end{minipage}%
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\hfill
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\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
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\begin{center}
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Top-input, right-output:
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\vspace{2mm}
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\begin{tikzpicture}[>=stealth,thick,baseline]
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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) {
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1 & 2 \\
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3 & 4 \\
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5 & 6 \\
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};
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\node[
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fit=(A-1-1)(A-1-1),
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inner xsep=0mm,inner ysep=3mm,
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label=above:5
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] (L) {};
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\draw[->, gray] (L.north) -- ([yshift=0mm]A-1-1.north);
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\node[
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fit=(A-1-2)(A-1-2),
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inner xsep=0mm,inner ysep=3mm,
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label=above:3
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] (R) {};
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\draw[->, gray] (R.north) -- ([yshift=0mm]A-1-2.north);
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\node[
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fit=(A-1-2)(A-1-2),
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inner xsep=8mm,inner ysep=0mm,
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label=right:{$5 + 6 = 11$}
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](Y) {};
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\draw[->, gray] ([xshift=3mm]A-1-2.east) -- (Y);
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\node[
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fit=(A-2-2)(A-2-2),
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inner xsep=8mm,inner ysep=0mm,
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label=right:{$15 + 12 = 27$}
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](H) {};
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\draw[->, gray] ([xshift=3mm]A-2-2.east) -- (H);
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\node[
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fit=(A-3-2)(A-3-2),
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inner xsep=8mm,inner ysep=0mm,
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label=right:{$25 + 18 = 43$}
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](N) {};
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\draw[->, gray] ([xshift=3mm]A-3-2.east) -- (N);
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\end{tikzpicture}
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\end{center}
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\end{minipage}%
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\vspace{2mm}
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This is only a model for intuition, though. \\
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Make sure you understand the dot product definition on the previous page.
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\vspace{5mm}
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\theorem{}<thebigtheorem>
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Any linear map $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as an $n \times m$ matrix. \\
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Conversely, every $n \times m$ matrix represents a liner map $T: \mathbb{R}^n \to \mathbb{R}^m$ \\
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\vspace{2mm}
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In other words, \textbf{matrices are linear transformations}.
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\problem{}<prooffwd>
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Show that the transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ defined by $T(v) = Av$ is linear. \\
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\hint{What is $A$? What is $v$? What are their sizes?}
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\vfill
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\problem{}<proofback>
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Show that any linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ can be written as $T(v) = Av$.
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\vfill
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\problem{}
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Consider the transformation $D: \mathbb{P}^3 \to \mathbb{P}^2$ defined by $D(p) = \frac{d}{dx}p$. \\
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Find a matrix that corresponds to $D$. \\
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\hint{$\mathbb{P}^3$ and $\mathbb{R}^4$ are isomorphic. How so?}
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\vfill
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\pagebreak
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\problem{}
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Show that the set of all linear maps is a vector space.
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\vfill
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\pagebreak |