EXERCISES FOR FINAL EXAM
I picked some exercises from our text book which should be helpful
for the final exam. The final exam will cover all material presented in
class. This contains Chapter 1 - 6, 7.2, 7.3, 7.4, 8.1, and part of 8.2
presented in class.
1. Part one: Basic conceptions
Chapter one P38 - P39 10, 21, 23, 25
Chapter two P98 - P100 4, 8, 30, 40
Chapter three P151 - P152 1, 2, 19, 34, 37
Chapter four P184 - P186 5, 11, 37, 42
Chapter five P247 - P248 7, 9, 16, 26, 39
Chapter six P292 - P293 4, 7, 12, 34, 35, 45
Chapter seven P365 - 366 8, 13, 30, 36, 46
Chapter eight P395 - P396 3, 5, 16, 20, 32, 48
Answer to Part I.
Chapter one P38 - P39
10. False, you may have one equation linearly dependent on others.
21. False. Note that even if the two matrices are similar, you can
not transform one to another by only row operation, or only column
operation. In fact, row operation can be viewed as multiply a matrix
from left; and column operation can be viewed as multiply a matrix
from right.
23. False.
1
2
EXERCISES FOR FINAL EXAM
25. False, you may have one equation linearly dependent on others.
Chapter two P98 - P100
4. True.
8. True.
30. True.
40. True. This can be verified directly using the formulas for projection matrices.
Chapter three P151 - P152
1. True.
2. False. Dimension of the image is at most 4.
19. True.
34. False. Similar matrix have the same determinants.
37. False. But the union of two spaces could be a subspace if one is
contained in the other.
Chapter four P184 - P186
5. True.
11. False. Note that sin x, cos x are in the kernel. In fact, they span
the kernel.
37. True.
42. False. The constant functions are in the kernel.
Chapter five P247 - P248
7. False. For T to be orthogonal, we need T~ei ’s form an orthonormal
basis.
EXERCISES FOR FINAL EXAM
3
9. True, since (A2 )−1 = (A2 )T .
16. True.
26. True.
39. True.
Chapter six P292 - P293
4. True. This holds for n × n matrices when n is even, but does not
hold for n is odd. (Why?)
7. True.
12. False.
34. False. This statement is true if A is an n × n matrix when n is
odd.
35. False.
45. True.
Chapter seven P365 - 366
8. True.
13. True.
30. False. This is in general not true, unless A, B have a same
eigen-basis.
36. False.
46. True.
Chapter eight P395 - P396
3. False.
4
EXERCISES FOR FINAL EXAM
5. True. This is equivalent to the definition of positive definite.
16. True.
20. False.
32. True, as the matrix in question is symmetric.
48. True, the sum of any quadratic forms is a quadratic form.
2. Part two: more exercises
This part consists of the exercises after each section, just like the
homework. I did not pick any exercises from Chapter one and two,
since the techniques from the first two chapters will be practiced in the
later chapters again and again . This does not mean that the first two
chapter is not important.
Chapter three
3.1/ 7, 15
3.2/ 15, 17
3.3/ 15, 31
3.4/ 3, 7, 15
Chapter four
4.1/ 1, 3, 29, 41
4.2/ 7, 22, 24, 39
4.3/ 19, 25, 39
Chapter five
5.1/ 6, 27
5.2/ 19, 29, 39
5.3/ 25, 27
5.4/ 5, 21, 23
5.5/ 3, 10
Chapter six
6.1/ 33, 37, 53
6.2/ 31, 34
6.3/ 3, 35, 39
EXERCISES FOR FINAL EXAM
Chapter seven
7.2/ 9, 11
7.3/ 9, 25, 34
7.4/ 7, 15, 29
Chapter eight
8.1/ 7, 9
8.2/ 3, 5, 11, 24
5