Added content

Advanced/Linear Algebra 101/parts/3 matrices.tex

@@ -0,0 +1,219 @@
+\section{Matrices}
+
+\definition{}
+A \textit{matrix} is a two-dimensional array of numbers: \\
+$$
+A =
+\begin{bmatrix}
+	1 & 2 & 3 \\
+	4 & 5 & 6
+\end{bmatrix}
+$$
+The above matrix has two rows and three columns. It is thus a $2 \times 3$ matrix.
+
+\definition{}<matvec>
+We can define the product of a matrix $A$ and a vector $v$:
+
+$$
+Av =
+\begin{bmatrix}
+	1 & 2 & 3 \\
+	4 & 5 & 6
+\end{bmatrix}
+\begin{bmatrix}
+	a \\ b \\ c
+\end{bmatrix}
+=
+\begin{bmatrix}
+	1a + 2b + 3c \\
+	4a + 5b + 6c
+\end{bmatrix}
+$$
+Note that each element of the resulting $2 \times 1$ matrix is the dot product of a row of $A$ with $v$:
+
+$$
+Av =
+\begin{bmatrix}
+	\text{---} a_1 \text{---} \\
+	\text{---} a_2 \text{---}
+\end{bmatrix}
+\begin{bmatrix}
+	| \\
+	v \\
+	| \\
+\end{bmatrix}
+=
+\begin{bmatrix}
+	r_1v \\
+	r_2v
+\end{bmatrix}
+$$
+
+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.
+
+
+\problem{}
+Say you multiply a size-$m$ vector by an $m \times n$ matrix. What is the size of your result?
+
+\vfill
+
+
+\problem{}
+Compute the following:
+
+$$
+\begin{bmatrix}
+	1 & 2 \\
+	3 & 4 \\
+	5 & 6
+\end{bmatrix}
+\begin{bmatrix}
+	5 \\ 3
+\end{bmatrix}
+$$
+
+\vfill
+\pagebreak
+
+\definition{}
+We also multiply a matrix by a matrix:
+
+$$
+AB =
+\begin{bmatrix}
+	1 & 2 \\
+	3 & 4
+\end{bmatrix}
+\begin{bmatrix}
+	10 & 20 \\
+	100 & 200
+\end{bmatrix}
+=
+\begin{bmatrix}
+	210 & 420 \\
+	430 & 860
+\end{bmatrix}
+$$
+Note each element of the resulting matrix is dot product of a row of $A$ and a column of $B$:
+
+$$
+AB =
+\begin{bmatrix}
+	\text{---} a_1 \text{---} \\
+	\text{---} a_2 \text{---}
+\end{bmatrix}
+\begin{bmatrix}
+	|	& | \\
+	v_1	& v_2 \\
+	|	& | \\
+\end{bmatrix}
+=
+\begin{bmatrix}
+	r_1v_1 & r_1v_2 \\
+	r_2v_1 & r_2vm_2 \\
+\end{bmatrix}
+$$
+
+\begin{center}
+\begin{tikzpicture}[>=stealth,thick,baseline]
+
+	\begin{scope}[layer = nodes]
+		\matrix[
+			matrix of math nodes,
+			left delimiter={[},
+			right delimiter={]}
+		] (A) at (0, 0){
+			1 & 2 \\
+			3 & 4 \\
+		};
+
+		\matrix[
+			matrix of math nodes,
+			left delimiter={[},
+			right delimiter={]}
+		] (B) at (2, 0) {
+			10 & 20 \\
+			100 & 200 \\
+		};
+
+		\node at (3.25, 0) {$=$};
+
+		\matrix[
+			matrix of math nodes,
+			left delimiter={[},
+			right delimiter={]}
+		] (C) at (4.5, 0) {
+			210 & 420 \\
+			430 & 860 \\
+		};
+	\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=-3mm,yshift=2mm]B-1-1) rectangle ([xshift=3mm,yshift=-2mm]B-1-2) {};
+
+	\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=-3mm,yshift=2mm]B-2-1) 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}
+\end{center}
+
+
+\problem{}
+Compute the following matrix product. \\
+
+$$
+\begin{bmatrix}
+	1 & 2 \\
+	3 & 4 \\
+	5 & 6
+\end{bmatrix}
+\begin{bmatrix}
+	9 & 8 & 7 \\
+	6 & 5 & 4
+\end{bmatrix}
+$$
+
+\vfill
+
+
+\problem{}
+Consider the following matrix product. \\
+Compute it or explain why you can't.
+
+$$
+\begin{bmatrix}
+	1 & 2 & 3 \\
+	4 & 5 & 6
+\end{bmatrix}
+\begin{bmatrix}
+	10 & 20 \\
+	30 & 40
+\end{bmatrix}
+$$
+
+\vfill
+
+
+\problem{}
+If $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, when does the product $AB$ exist?
+
+
+\vfill
+\pagebreak
+
+
+\problem{}
+Look back to \ref{matvec}. \\
+Convince yourself that vectors are matrices. \\
+
+Can you multiply a matrix by a vector, as in $vA$? \\
+How does the dot prouduct relate to matrix multiplication? (transpose)
+
+\vfill
+\pagebreak