Linear Algebra with Applications, 10th Edition

Chapter Test A

1. 1. Hint: Make up an example using linearly independent vectors x and y.

2. 2. Hint: If $\left|{\mathbf{\text{x}}}^T\mathbf{\text{y}}\right|=1$, what can you conclude about the angle between the vectors?

3. 3. Hint: Look at Exercise 14 of Section 5.1. If you can find vectors ${\mathbf{\text{x}}}_1,{\mathbf{\text{x}}}_2,{\mathbf{\text{x}}}_3$ such that ${\mathbf{\text{x}}}_1\perp {\mathbf{\text{x}}}_2$ and ${\mathbf{\text{x}}}_2\perp {\mathbf{\text{x}}}_3$, but ${\mathbf{\text{x}}}_1$ is not orthogonal to ${\mathbf{\text{x}}}_3$, then consider the subspaces

   $$S_1={\text{Span}(\text{x}}_1),S_2={\text{Span}(\text{x}}_2),S_3={\text{Span}(\text{x}}_3)$$

4. 4. Hint: If $A^T\mathbf{\text{y}}=\mathbf{0}$, then y is in $N\left(A^T\right)$.

5. 5. Hint: The matrices A and $A^{T}A$ have the same rank. (See Exercise 13 of Section 5.2.)

6. 6. Hint: The least squares problem will not have a unique solution but that doesn’t imply` the projection is not unique. See Theorem 5.3.1.

7. 7. Hint: If A is $m\times n$ and $N\left(A\right)=\left\{\mathbf{0}\right\}$, then what can you conclude about the rank of A?

8. 8. Hint: Check to ...