Chapter Test A

1. Hint: Make up an example using linearly independent vectors x and y.
2. Hint: If $| x^{T} y | = 1$ $| x^{T} y | = 1$ , what can you conclude about the angle between the vectors?
3. Hint: Look at Exercise 14 of Section 5.1. If you can find vectors $x_{1}, x_{2}, x_{3}$ $x_{1}, x_{2}, x_{3}$ such that $x_{1} ⊥ x_{2}$ $x_{1} ⊥ x_{2}$ and $x_{2} ⊥ x_{3}$ $x_{2} ⊥ x_{3}$ , but $x_{1}$ $x_{1}$ is not orthogonal to $x_{3}$ $x_{3}$ , then consider the subspaces

$S_{1} = {Span (x}_{1}), S_{2} = {Span (x}_{2}), S_{3} = {Span (x}_{3})$ $S_{1} = {Span (x}_{1}), S_{2} = {Span (x}_{2}), S_{3} = {Span (x}_{3})$
4. Hint: If $A^{T} y = 0$ $A^{T} y = 0$ , then y is in $N (A^{T})$ $N (A^{T})$ .
5. Hint: The matrices A and $A^{T} A$ $A^{T} A$ have the same rank. (See Exercise 13 of Section 5.2.)
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. Hint: If A is $m \times n$ $m \times n$ and $N (A) = {0}$ $N (A) = {0}$ , then what can you conclude about the rank of A?
8. Hint: Check to see if ${(Q_{1} Q_{2})}^{T} (Q_{1} Q_{2}) = I$ ${(Q_{1} Q_{2})}^{T} (Q_{1} Q_{2}) = I$ .
9. Hint: How is the (i, j) entry of $U^{T} U$ $U^{T} U$ determined?
10. Hint: Make up some examples (with $k < n$ $k < n$ ) to see if the statement is true.

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