Linear Algebra with Applications, 10th Edition

Chapter Test A

1. 1. Hint: If L is represented by a matrix A, then $L\left(\mathbf{x}\right)=A\mathbf{\text{x}}$ for each $\mathbf{x}\in R^2\mathrm{.}$ Does $A{\mathbf{\text{x}}}_1=A{\mathbf{\text{x}}}_2$ necessarily imply that ${\mathbf{x}}_1={\mathbf{x}}_2?$

2. 2. Hint: Show that

   $$\left(L_1+L_2\right)\left(c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2\right)=c_1\left(L_1+L_2\right)\left({\mathbf{v}}_1\right)+c_2\left(L_1+L_2\right)\left({\mathbf{v}}_2\right)$$

3. 4. Hint: Hint. Make up some vectors. For each vector x you make up, test to see if $L_1\left(\mathbf{\text{x}}\right)=L_2\left(\mathbf{\text{x}}\right)\mathrm{.}$

4. 5. Hint: Check to see if the homogeneous coordinate system is closed under the operations of scalar multiplication and vector addition.

5. 6. Hint: Check to see if

   $$L^2\left(c_1{\mathbf{\text{x}}}_1+c_2{\mathbf{\text{x}}}_2\right)=c_1{L}^2\left({\mathbf{\text{x}}}_1\right)+c_2{L}^2\left({\mathbf{\text{x}}}_2\right)$$

6. 7. Hint: Let $L_1$ and $L_2$ be linear transformations that are both represented by the same matrix A with respect to the ordered basis E. Will $L_1\left(\mathbf{\text{x}}\right)=L_2\left(\mathbf{\text{x}}\right)$ for every $\mathbf{\text{x}}\in R^n?$

7. 8. Hint: See Theorem 4.3.1.

8. 9. Hint: If

   $$A=X^{-1}\mathit{BX}\mathrm{and}B=Y^{-1}\mathit{CY}$$

   can you ...