Matrix-vector products

Definition: The matrix-vector product $Ax$ is the product of an $m \times n$ matrix $A$ and a column vector $x \in \mathbb{R}^n$ (or $\mathbb{C}^n$), resulting in a column vector $b \in \mathbb{R}^m$ (or $\mathbb{C}^m$).

$$b = Ax = \begin{pmatrix} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n \end{pmatrix}$$

Equivalently, $Ax$ is a linear combination of the columns of $A$ weighted by the entries of $x$:

$$Ax = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \cdots + x_n \mathbf{a}_n$$

where $\mathbf{a}_j$ is the $j$-th column of $A$.

Example:
$$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix} = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$$