Matrix Product as Linear Transformation

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Theorem

Let:

\(\ds \mathbf A_{m \times n}\) \(=\) \(\ds \begin {bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end {bmatrix}\)
\(\ds \mathbf x_{n \times 1}\) \(=\) \(\ds \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix}\)
\(\ds \mathbf y_{n \times 1}\) \(=\) \(\ds \begin {bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end {bmatrix}\)

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.

$\blacksquare$


Also see


Sources

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