Linear Algebra with Applications, 10th Edition

4.2 Matrix Representations of Linear Transformations

In Section 4.1, it was shown that each $m\times n$ matrix A defines a linear transformation $L_A$ from ${\mathrm{ℝ}}^n$ to ${\mathrm{ℝ}}^m$, where

$$L_A\left(\mathbf{x}\right)=A\mathbf{x}$$

for each $\mathbf{\text{x}}\in {\mathrm{ℝ}}^n$. In this section, we will see that, for each linear transformation L mapping ${\mathrm{ℝ}}^n$ into ${\mathrm{ℝ}}^m$, there is an $m\times n$ matrix A such that

$$L\left(\mathbf{x}\right)=A\mathbf{x}$$

We will also see how any linear transformation between finite dimensional spaces can be represented by a matrix.

Theorem 4.2.1

If L is a linear transformation mapping ${\mathrm{ℝ}}^n$ into ${\mathrm{ℝ}}^m$, there is an $m\times n$ matrix A such that

$$L\left(\mathbf{x}\right)=A\mathbf{x}$$

for each $\mathbf{x}\in {\mathrm{ℝ}}^n$. In fact, the jth column vector of A is given by

$$\begin{array}{cc}{\mathbf{\text{a}}}_j=L({\mathbf{\text{e}}}_j)& j=1,2,\mathrm{\dots },n\end{array}$$

Proof

For $j=1,\mathrm{\dots },n$, define

$$\begin{array}{c}{\mathbf{\text{a}}}_j=L({\mathbf{\text{e}}}_j)\end{array}$$

and let

$$A=\left(a_{ij}\right)=\left({\mathbf{a}}_1,{\mathbf{a}}_2,\mathrm{\dots },{\mathbf{a}}_n\right)$$

If

$$\mathbf{x}=x_1{\mathbf{e}}_1+x_2{\mathbf{e}}_2+\mathrm{\dots }+x_n{\mathbf{e}}_n$$

is an arbitrary element ...