Linear Transformation as Matrix Product

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Theorem

Let $T: \R^n \to \R^m, \mathbf x \mapsto T\left({\mathbf x}\right)$ be a linear transformation.

Then:

$T \left({\mathbf x}\right) = \mathbf A_T \mathbf x$

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

$\mathbf A_T = \begin{bmatrix} T \left({\mathbf e_1}\right) & T \left({\mathbf e_2 }\right) & \cdots & T \left({\mathbf e_n}\right)\end{bmatrix}$

where $\left({\mathbf e_1, \mathbf e_2, \cdots, \mathbf e_n}\right)$ 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:

\(\displaystyle \mathbf x_{n \times 1}\) \(=\) \(\displaystyle \mathbf I_n \mathbf x_{n \times 1}\) Definition of Left Identity
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \\ \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\) Unit Matrix is Identity:Lemma
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} \mathbf e_1 & \mathbf e_2 & \cdots & \mathbf e_n \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\) Definition of Standard Ordered Basis
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \mathbf e_i x_i\) Definition of Matrix Multiplication
\(\displaystyle T \left({\mathbf x}\right)\) \(=\) \(\displaystyle T \left({\sum_{i \mathop =1}^n \mathbf e_i x_i}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n T \left({\mathbf e_i}\right)x_i\) Definition of Linear Transformation
\(\displaystyle \) \(=\) \(\displaystyle \begin{bmatrix} T \left({\mathbf e_1}\right) & T \left({\mathbf e_2}\right) & \cdots & T \left({\mathbf e_n}\right) \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}\) Definition of Matrix Multiplication

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

$\blacksquare$


Also see


Sources