Linear Algebra with Applications, 10th Edition

Section 4.3 Exercises

1. For each of the following linear operators L on ${ℝ}^2$, determine the matrix A representing L with respect to $\left\{{\mathbf{e}}_1,{\mathbf{e}}_2\right\}$ (see Exercise 1 of Section 1.2) and the matrix B representing L with respect to $\left\{{\mathbf{u}}_1={\left(1,1\right)}^T,{\mathbf{u}}_2={\left(-1,1\right)}^T\right\}$:

   1. $L\left(\mathbf{x}\right)={\left(-x_1,x_2\right)}^T$

   2. $L\left(\mathbf{x}\right)=-\mathbf{x}$

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

   4. $L\left(\mathbf{x}\right)=\frac{1}{2}\mathbf{x}$

   5. $L\left(\mathbf{x}\right)=x_2{\mathbf{e}}_2$

2. Let $\left\{{\mathbf{u}}_1,{\mathbf{u}}_2\right\}$ and $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ be ordered bases for ${ℝ}^2$, where

   $$\begin{array}{cc}{\mathbf{u}}_1=\left[\begin{array}{c}1\\ 1\end{array}\right],& {\mathbf{u}}_2=\left[\begin{array}{c}-1\\ 1\end{array}\right]\end{array}$$

   and

   $$\begin{array}{cc}{\mathbf{\text{v}}}_1=\left[\begin{array}{c}2\\ 1\end{array}\right],& {\mathbf{v}}_2=\left[\begin{array}{c}1\\ 0\end{array}\right]\end{array}$$

   Let L be the linear transformation defined by

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

   and let B be the matrix representing L with respect to $\left\{{\mathbf{u}}_1,{\mathbf{u}}_2\right\}$ [from Exercise 1(a)].

   1. Find the transition matrix S corresponding to the change of basis from $\left\{{\mathbf{u}}_1,{\mathbf{u}}_2\right\}$ to $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$.

   2. Find the matrix A representing L with respect to $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$ by computing $SB{S}^{-1}$.

   3. Verify that

      $$\begin{array}{cc}\hfill L\left({\mathbf{v}}_1\right)& =a_{11}{\mathbf{v}}_1\hfill \end{array}$$