Linear Algebra with Applications, 10th Edition

Chapter Test A

1. 1. Hint: See the discussion at the beginning of Section3.2 .

2. 2. Hint: If x is a vector in ${ℝ}^2,$ then it has only 2 entries.

3. 3. Hint: A two dimensional subspace of ${ℝ}^3$ corresponds to a plane through the origin in 3-space.

4. 4. Hint: See Exercise22 of Section3.2 .

5. 5. Hint: See Exercise21 of Section3.2 .

6. 6. Hint: See Theorem 3.4.3.

7. 7. Hint: Is it possible for three vectors ${\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,{\mathbf{\text{x}}}_3$ to span the vector space ${ℝ}^2?$

8. 8. Hint: If

   $$\text{S}\text{pan}({\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,\mathrm{...},{\mathbf{\text{x}}}_k)=\text{S}\text{pan}({\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,\mathrm{...},{\mathbf{\text{x}}}_{k-1})$$

   then ${\mathbf{\text{x}}}_k$ must be in $\text{S}\text{pan}({\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,\mathrm{...},{\mathbf{\text{x}}}_{k-1})\mathrm{.}$

9. 9. Hint: See Theorem 3.6.5.

10. 10. Hint: Use the Rank-Nullity theorem to determine the nullities of the matrices.

11. 12. Hint: Look at some examples.

12. 13. Hint: Show that if ${\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,\mathrm{...},{\mathbf{\text{x}}}_k,{\mathbf{\text{x}}}_{k+1}$ were linearly dependent and ...