Matrix Product as Linear Transformation

Theorem
Let:


 * $ \mathbf A_{m \times n} = \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}$


 * $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$


 * $\mathbf y_{n \times 1} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$

be matrices where each column is an element of a real vector space.

Let $T$ be the mapping:


 * $T: \R^m \to \R^n, \mathbf x \mapsto \mathbf A \mathbf x$

Then $T$ is a linear transformation.

Proof
From Matrix Multiplication is Homogeneous of Degree $1$:
 * $\forall \lambda \in \mathbb F \in \set {\R, \C}: \mathbf A \paren {\lambda \mathbf x} = \lambda \paren {\mathbf A \mathbf x}$

From Matrix Multiplication Distributes over Matrix Addition:
 * $\forall \mathbf x, \mathbf y \in \R^m: \mathbf A \paren {\mathbf x + \mathbf y} = \mathbf A \mathbf x + \mathbf A \mathbf y$

Hence the result, from the definition of linear transformation.

Also see

 * Linear Transformation as Matrix Product