Linear Algebra with Applications, 10th Edition

4.1

1. 1. 

   1. (a) reflection about $x_2$ axis;

   2. (b) reflection about the origin;

   3. (c) reflection about the line $x_2=x_1$;

   4. (d) the length of the vector is halved;

   5. (e) projection onto $x_2$ axis

2. 4. ${(7,18)}^T$

3. 5. All except (c) are linear transformations from $R^3$ into $R^2$.

4. 6. (b) and (c) are linear transformations from $R^2$ into $R^3$.

5. 7. (a), (b), and (d) are linear transformations.

6. 9. (a) and (c) are linear transformations from $P_2$ into $P_3$.

7. 10. $L(e^x)=e^x-1$ and $L(x^2)=x^3/3$.

8. 11. (a) and (c) are linear transformations from $C[0,1]$ into $R^1$.

9. 13. Hint: If v is any element of V then

   $$\mathbf{v}={\alpha }_1{\mathbf{v}}_1+{\alpha }_2{\mathbf{v}}_2+\mathrm{...}+{\alpha }_n{\mathbf{v}}_n$$

10. 17. 

    1. (a) $\text{ker}(L)=\{\mathbf{0}\},L(R^3)=R^3$;

    2. (c) $\begin{array}{c}\text{ker}(L)=\text{Span}({\mathbf{e}}_2,{\mathbf{e}}_3)\hfill \end{array},\begin{array}{c}L(R^3)=\text{Span}({(1,1,1)}^T)\hfill \end{array}$

11. 18. 

    1. (a) $\begin{array}{c}L(S)=\text{Span}({\mathbf{e}}_2,{\mathbf{e}}_3)\hfill \end{array}$;

    2. (b) $\begin{array}{c}L(S)=\text{Span}({\mathbf{e}}_1,{\mathbf{e}}_2)\hfill \end{array}$

12. 19. 

    1. (a) $\text{ker}(L)=P_1,L(P_3)=\text{Span}(x^2,x)$;

    2. (c) $\text{ker}$