Linear Algebra with Applications, 10th Edition

Section 4.2 Exercises

1. Refer to Exercise 1 of Section 4.1. For each linear transformation L, find the standard matrix representation of L.

2. For each of the following linear transformations L mapping ${ℝ}^3$ into ${ℝ}^2$, find a matrix A such that L $L\left(\mathbf{x}\right)=A\mathbf{x}$ for every x in ${ℝ}^3$:

   1. $L({(x_1,x_2,x_3)}^T={(x_1+x_2,0)}^T)$

   2. $L({(x_1,x_2,x_3)}^T)={(x_1+x_2)}^T$

   3. $L({(x_1,x_2,x_3)}^T)={(x_2-x_1,x_3-x_2)}^T$

3. For each of the following linear operators L on ${ℝ}^3$, find a matrix A such that $L\left(\mathbf{x}\right)=A\mathbf{x}$ for every x in ${ℝ}^3$:

   1. $L\left({\left(x_1,x_2,x_3\right)}^T\right)={\left(x_3,x_2,x_1\right)}^T$

   2. $L\left({\left(x_1,x_2,x_3\right)}^T\right)={\left(x_1,x_1+x_2,x_1+x_2+x_3\right)}^T$

   3. $L\left({\left(x_1,x_2,x_3\right)}^T\right)={\left(2x_3,x_2+3x_1,2x_1-x_3\right)}^T$

4. Let L be the linear operator on ${ℝ}^3$ defined by

   $$L\left(\text{x}\right)=\left[\begin{array}{c}2x_1-x_2-x_3\\ 2x_2-x_1-x_3\\ 2x_3-x_1-x_2\end{array}\right]$$

   Determine the standard matrix representation A of L, and use A to find L (x) for each of the following ...