Page 146 Chapter 3 True False Questions.

1. The image of a 3x4 matrix is a subspace of R4?

False. It is a subspace of R3.

2. The span of vectors V1, V2, …,Vn consists of all linear combinations of vectors V1, V2, …, Vn.

True. That is the definition of the span.

3. If V1, V2, …, Vn are linearly independent vectors in Rn, then they must form a basis of Rn.

True: n linearly independent vectors in a space of dimension n form a basis.

4. There is a 5x4 matrix whose image consists of all of R5.

False. It takes at least 5 vectors to span

all of R5.

5. The kernel of any invertible matrix consists of the zero vector only.

True. AX = 0 implies X = 0 when A is

invertible.

6. The identity matrix In is similar to all invertible nxn matrices.

False. The identity matrix is similar only to

itself. A-1 I A = I for all invertible matrices

A.

7. If 2 U + 3 V + 4 W = 5U + 6 V + 7 W, then vectors U, V, W must be linearly dependent.

True. In fact 3U+3V+3W = 0.

8. The column vectors of a 5x4 matrix must be linearly dependent.

False. | 1 0 0 0 |

| 0 1 0 0 |

| 0 0 1 0 |

| 0 0 0 1 |

| 0 0 0 0 |

is an example where they are linearly independent.

9. If V1, V2, …, Vn and W1, W2, …, Wm are any two bases of a subspace V of R10, then n must equal m.

True. Any two bases of the same vector space have the same number of vectors.

10. If A is a 5x6 matrix of rank 4, then the nullity of A is 1.

False. The rank plus the nullity is the number of columns. Thus the nullity would be 2.

11. If the kernel of a matrix A consists of the zero vector only, then the column vectors of A must be linearly independent.

True. Since the kernel is zero, the columns of A must be linearly independent.

12. If the image of an nxn matrix A is all of Rn, then A must be invertible.

True. Since the columns span Rn , the matrix must have a right inverse. Since it

is square, it must be invertible.

13. If vectors V1, V2, …, Vn span R4 then n must be equal to 4.

False. It could be 4 or larger than 4.

14. If vectors U, V, and W are in a subspace V of Rn, then 2 U – 3 V + 4 W must be in V as well.

True. A subspace is closed under addition and scalar multiplication.

15. If matrix A is similar to matrix B, and B is similar to C, then C must be similar to A.

True. P-1AP = B

Q-1BQ = C

Q-1P-1APQ = C

A = PQCQ-1P-1

A = (Q-1P-1)-1 C (Q-1P-1)

16. If a subspace V of Rn contains none of the standard vectors E1, E2, …, En, then V consists of the zero vector only.

| c |

False. The space | c | of R3 is a

| c |

counter example.

17. If vectors V1, V2, V3, V4 are linearly independent, then vectors V1, V2, V3 must be linearly independent as well.

True. Any dependence relation among V1, V2, V3 can be made into a dependence relation for V1, V2, V3, V4 by adding a zero coefficient to V4.

| a |

18. The vectors of the form | b |

| 0 |

| a |

(where a and b are arbitrary real numbers) form a subspace of R4.

True. This is closed under addition and scalar multiplication.

19. Matrix | 1 0 | is similar to | 0 1 |.

| 0 -1 | | 1 0 |

True. |1/2 -1/2 | | 1 0| |1/2 -1/2 |-1 = | 0 1 |

|1/2 1/2 | | 0 -1| |1/2 1/2| | 1 0 |

| 1 | | 2 | | 3 |

20. Vectors | 0 |, | 1 |, | 2 | form a basis of R3.

| 0 | | 0 | | 1 |

| 1 | | 2 | | 3 | |a+2b+3c|

True. a| 0 |+b| 1 |+c| 2 | = | b+2c |

| 0 | | 0 | | 1 | | c |

For the dependence relation to equal zero,

we must have c = 0, then b=0, then a=0. Thus

the three vectors are linearly independent and

must be a basis of R3.

21. Matrix | 0 1 | is similar to | 0 0 |.

| 0 0 | | 0 1 |

False. The first matrix squares to zero. The

second matrix does not square to zero. They

cannot be similar.

22. These vectors are linearly independent.

| 1 | | 5 | | 9 | | 5 | | 1 |

| 2 | | 6 | | 8 | | 4 | | 0 |

| 3 | | 7 | | 7 | | 3 | |-1 |

| 4 | | 8 | | 6 | | 2 | |-2 |

False. They are five vectors in a space of

dimension 4. They must be linearly dependent.

23. If a subspace V of R3 contains the standard vectors E1, E2, E3, then V must be R3.

True. Clearly everything is a linear combination of E1, E2, and E3.

24. If a 2x2 matrix P represents the orthogonal projection onto a line in R2, then P must be similar to matrix | 1 0 |.

| 0 0 |

True. Use one basis vector along the line

things are projected onto, and put the other

basis vector along the line perpendicular to the first.

25. If A and B are nxn matrices, and vector V is in the kernel of both A and B, then V must be in the kernel of matrix AB as well.

True. In fact we did not even need V to be in the kernel of A. If V is in the kernel of B, then V is in the kernel of AB.

26. If two nonzero vectors are linearly dependent, then each of them is a scalar multiple of the other.

True. The dependence relation aV+bW = 0

has to have both a and b nonzero. Then

V = -b/a W and W = -a/b V.

27. If V1, V2, V3 are any three vectors in R3, then there must be a linear transformation T from R3 to R3 such that T(V1) = E1, T(V2) = E2, and T(V3) = E3.

False. You can do this when they are independent. You cannot do it when they are dependent.

28. If vectors U, V, W are linearly dependent, then vector W must be a linear combination of U and V.

False. Let U = V = 0 and W = E3.

29. If A and B are invertible nxn matrices, then AB is similar to BA.

True. A-1(AB)A = BA

30. If A is an invertible nxn matrix, then the

kernels of A and A-1 must be equal.

True. In fact the kernels of A and A-1 are both just 0.

31. If V is any three-dimensional subspace of R5 then V has infinitely many bases.

True. If V1, V2, V3 is one basis, then

V1+kV2, V2, V3 is another basis for

each integer k.

32. Matrix In is similar to 2 In.

False. In is similar to only itself.

33. If AB = 0 for two 2x2 matrices A and B, then BA must be the zero matrix as well.

False. | 0 0 | | 0 0 | = | 0 0 |

| 1 0 | | 0 1 | | 0 0 |

| 0 0 | | 0 0 | = | 0 0 |

| 0 1 | | 1 0 | | 1 0 |

34. If A and B are nxn matrices, and V is in the image of both A and B, then V must be in the image of matrix A+B as well.

False. Consider B = -A. Then

A+B = 0 yet A and B have the same image.

35. If V and W are subspaces of Rn, then their union VuW must be a subspace of Rn as well.

False. V = | c | W = | 0 |.

| 0 | | d |

Then VuW is not closed under addition since

| c | is not in the union.

| d |

36. If the kernel of a 5x4 matrix A consists of the zero vector only and if AV = AW for two vectors V and W in R4, then vectors V and W must be equal.

True. Since A(V-W) = 0, V-W = 0 and so V=W.

37. If V1, V2, …, Vn and W1, W2, …, Wn are two bases of Rn, then there is a linear transformation T from Rn to Rn such that T(V1) = W1, T(V2) = W2, …, T(Vn) = Wn.

True. You can map a basis anywhere.

38. If matrix A represents a rotation through Pi/2 and matrix B rotation through Pi/4, then A is similar to B.

False. A = | 0 -1 | B = | 1/Sqrt[2] -1/Sqrt[2] |

| 1 0 | | 1/Sqrt[2] 1/Sqrt[2] |

A4 = I and B4 =/= I. They cannot be similar.

39. R2 is a subspace of R3.

False. There are subspaces of R3 of dimension 2, but the vectors in them are all three tuples, not 2 tuples.

40. If an nxn matrix A is similar to matrix B, then A + 7In must be similar to B + 7 In.

True. If P-1AP = B then

P-1(A+7In)P = P-1AP + 7 P-1 In P = B + 7 In

41. There is a 2x2 matrix A such that im(A) = ker(A).

True. | 0 1 | is one such matrix.

| 0 0 |

42. If two nxn matrices A and B have the same rank, then they must be similar.

False. | 1 0 | and | 0 1 | both have rank

| 0 0 | | 0 0 |

one, but are not similar.

43. If A is similar to B, and A is invertible, then B must be invertible as well.

True. If P-1 A P = B then

P-1 A-1 P = B-1

44. If A2 = 0 for a 10x10 matrix A, then the inequality rank(A) <= 5 must hold.

True. 10 = rank(A) + nullity(A)

Since A is contained in the null space of A,

10 >= 2 rank(A). So rank(A) <= 5.

45. For every subspace V of R3 there is a 3x3 matrix A such that V = im(A).

True. Just pick 3 vectors which span V. Use

these as the columns of the matrix.

46. There is a nonzero 2x2 matrix A that is similar to 2A.

True. | 2 0 | | 0 1 | | ½ 0 | = | 0 2 |

| 0 1 | | 0 0 | | 0 1 | | 0 0 |

47. If the 2x2 matrix R represents the reflection across a line in R2, then R must be similar to the matrix | 0 1 |.

| 1 0 |

True. Use the basis

| /

_____|/_______

|\

| \

48. If A is similar to B, then there is one and only

one invertible matrix S such that S-1 A S = B.

False. (A-1S)-1 A (A-1 S) will also work.

49. If the kernel of a 5x4 matrix A consists of the zero vector alone, and if AB = AC for two 4x5 matrices B and C, then the matrices B and C must be equal.

True. A(B-C) = 0 so B-C = 0 and so B=C.

50. If A is any nxn matrix such that A2 = A, then the image of A and the kernel of A have only the zero vector in common.

True. If A(AV) = 0, that is, if AV is in the image of A and also in the kernel of A, then

0 = A2V = AV.

51. There is a 2x2 matrix A such that A2 =/= 0 and A3 = 0.

False. If A2 V =/= 0, and A3 V = 0, then

V, AV, A2V must be linearly independent.

This is impossible in R2.