Linear Transformation as Matrix Product

Theorem
Let $T: \R^n \to \R^m, \mathbf x \mapsto \map T {\mathbf x}$ be a linear transformation.

Then:
 * $\map T {\mathbf x} = \mathbf A_T \mathbf x$

where $\mathbf A_T$ is the $m \times n$ matrix defined as:


 * $\mathbf A_T = \begin {bmatrix} \map T {\mathbf e_1} & \map T {\mathbf e_2} & \cdots & \map T {\mathbf e_n} \end {bmatrix}$

where $\tuple {\mathbf e_1, \mathbf e_2, \cdots, \mathbf e_n}$ is the standard ordered basis of $\R^n$.

Proof
Let $\mathbf x = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix}$.

Let $\mathbf I_n$ be the unit matrix of order $n$.

Then:

That $\mathbf A_T$ is $m \times n$ follows from each $\map T {\mathbf e_i}$ being an element of $\R^m$ and thus having $m$ rows.

Also see

 * Matrix Product as Linear Transformation