Linear Algebra with Applications, 10th Edition

3.2

1. 1. (a) and (c) are subspaces; (b), (d), and (e) are not.

2. 2. (b) and (c) are subspaces; (a) and (d) are not.

3. 3. (a), (c), (e), and (f) are subspaces; (b), (d), and (g) are not.

4. 4. 

   1. (a) $\{{(0,0)}^T\}$;

   2. (b) $\text{Span}({(-2,1,0,0)}^T,{(3,0,1,0)}^T)$;

   3. (c) $\text{Span}({(1,1,1)}^T)$;

   4. (d) $\text{Span}({(-5,0,-3,1)}^T,(-1,1,0,0{)}^T)$

5. 5. Only the set in part (c) is a subspace of $P_4$.

6. 6. (a), (b), and (d) are subspaces.

7. 10. 

   1. (b) Hint: If $AB\ne BA,$ then is $A\left(cB\right)\ne \left(cB\right)A$ for all scalars c?

8. 11. (a), (c), and (e) are spanning sets.

9. 12. (a) and (b) are spanning sets.

10. 13.  

    1. (a) Hint: $\mathbf{\text{x}}\in {\text{Span}(\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2)$ if and only if there exist scalars $c_1$ and $c_2$ such that

       $$c_1{\mathbf{\text{x}}}_1+c_2{\mathbf{\text{x}}}_2=\mathbf{\text{x}}$$

       Thus, $\mathbf{\text{x}}\in {\text{Span}(\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2)$ if and only if the linear system $X\mathbf{\text{c}}=\mathbf{\text{x}}$ is consistent.

11. 14. Hint: Is it possible to make any conclusions as to whether ...