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 a member of a real vector space.

be matrices where each column is a member 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

 * $\forall \lambda \in \mathbb F \in \left \{ \R, \C \right \}, \mathbf A \left({\lambda \mathbf x }\right) = \lambda \left({\mathbf A \mathbf x}\right)$, from Matrix Multiplication Homogeneous of Degree $1$.


 * $\forall \mathbf x, \mathbf y \in \R^m: \mathbf A \left({\mathbf x + \mathbf y}\right) = \mathbf A \mathbf x + \mathbf A \mathbf y$, from Matrix Multiplication Distributes over Matrix Addition.

Hence the result, from the definition of a linear transformation.

Also see

 * Linear Transformation as Matrix Product