Linear Algebra with Applications, 10th Edition

4.3 Similarity

If L is a linear operator on an n-dimensional vector space V, the matrix representation of L will depend on the ordered basis chosen for V. By using different bases, it is possible to represent L by different $n\times n$ matrices. In this section, we consider different matrix representations of linear operators and characterize the relationship between matrices representing the same linear operator.

Let us begin by considering an example in ${\mathrm{ℝ}}^2$. Let L be the linear transformation mapping ${\mathrm{ℝ}}^2$ into itself defined by

$$L\left(\mathbf{x}\right)={\left(2x_1,x_1+x_2\right)}^T$$

Since

$$\begin{array}{cc}L\left({\mathbf{e}}_1\right)=\left[\begin{array}{c}2\\ 1\end{array}\right]& \begin{array}{cc}\text{and}& L\left({\mathbf{e}}_2\right)=\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}\end{array}$$

it follows that the matrix representing L with respect to $\left\{{\mathbf{e}}_1,{\mathbf{e}}_2\right\}$ is

$$A=\left[\begin{array}{cc}2& 0\\ 1& 1\end{array}\right]$$

If we use a different basis for ${ℝ}^2$, the matrix representation of L will change. If, for example, ...