We can “multiply” a matrix and a vector together. This type of matrix-vector product represents the result of a linear transformation on a (geometric) vector, and it is only defined for cases where the number of columns in the matrix equals the number of rows in the vector. For the 2x2 case, the algebraic definition is quite simple:
\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a x + b y \\ c x + d y \end{bmatrix} \]
In the 3x3 case:
\[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} a x + b y + c z \\ d x + e y + f z \\ g x + h y + i z \end{bmatrix} \]
Of course the matrix does not need to be square. When it is not square, the input vector and the output vector have different dimensions. In that case, the matrix represents a linear transformation between two different spaces. For example, for a 2x3 matrix:
\[ \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} a x + b y + c z \\ d x + e y + f z \end{bmatrix} \]
In general the matrix-vector between an \(m \times n\) matrix and an \(n \times 1\) vector is defined to be
\[ \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \cdot \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n \\ a_{21} x_1 + a_{22} x_2 + \cdots + a_{2n} x_n \\ \vdots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n \end{bmatrix} \]
Note that the result is an \(m \times 1\) vector.