1. Page 146 Chapter 3 True False Questions. 1. The image of a 3x4 matrix is a subspace of R 4 ? False. It is a subspace of R 3.

2. 2. The span of vectors V 1, V 2, …,V n consists of all linear combinations of vectors V 1, V 2, …, V n. True. That is the definition of the span.

3. 3. If V 1, V 2, …, V n are linearly independent vectors in R n, then they must form a basis of R n. True: n linearly independent vectors in a space of dimension n form a basis.

4. 4. There is a 5x4 matrix whose image consists of all of R 5. False. It takes at least 5 vectors to span all of R 5.

5. 5. The kernel of any invertible matrix consists of the zero vector only. True. AX = 0 implies X = 0 when A is invertible.

6. 6. The identity matrix I n 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. 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. 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. 9. If V 1, V 2, …, V n and W 1, W 2, …, W m are any two bases of a subspace V of R 10, then n must equal m. True. Any two bases of the same vector space have the same number of vectors.

10. 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. 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. 12. If the image of an nxn matrix A is all of R n, then A must be invertible. True. Since the columns span R n, the matrix must have a right inverse. Since it is square, it must be invertible.

13. 13. If vectors V 1, V 2, …, V n span R 4 then n must be equal to 4. False. It could be 4 or larger than 4.

14. 14. If vectors U, V, and W are in a subspace V of R n, then 2 U – 3 V + 4 W must be in V as well. True. A subspace is closed under addition and scalar multiplication.

15. 15. If matrix A is similar to matrix B, and B is similar to C, then C must be similar to A. True. P -1 AP = B Q -1 BQ = C Q -1 P -1 APQ = C A = PQCQ -1 P -1 A = (Q -1 P -1 ) -1 C (Q -1 P -1 )

16. 16. If a subspace V of R n contains none of the standard vectors E 1, E 2, …, E n, then V consists of the zero vector only. | c | False. The space | c | of R 3 is a | c | counter example.

17. 17. If vectors V 1, V 2, V 3, V 4 are linearly independent, then vectors V 1, V 2, V 3 must be linearly independent as well. True. Any dependence relation among V 1, V 2, V 3 can be made into a dependence relation for V 1, V 2, V 3, V 4 by adding a zero coefficient to V 4.

18. | a | 18. The vectors of the form | b | | 0 | | a | (where a and b are arbitrary real numbers) form a subspace of R 4. True. This is closed under addition and scalar multiplication.

19. 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 |

20. | 1 | | 2 | | 3 | 20. Vectors | 0 |, | 1 |, | 2 | form a basis of R 3. | 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 R 3.

21. 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. 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. 23. If a subspace V of R 3 contains the standard vectors E 1, E 2, E 3, then V must be R 3. True. Clearly everything is a linear combination of E 1, E 2, and E 3.

24. 24. If a 2x2 matrix P represents the orthogonal projection onto a line in R 2, 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. 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. 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. 27. If V 1, V 2, V 3 are any three vectors in R 3, then there must be a linear transformation T from R 3 to R 3 such that T(V 1 ) = E 1, T(V 2 ) = E 2, and T(V 3 ) = E 3. False. You can do this when they are independent. You cannot do it when they are dependent.

28. 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 = E 3.

29. 29. If A and B are invertible nxn matrices, then AB is similar to BA. True. A -1 (AB)A = BA

30. 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. 31. If V is any three-dimensional subspace of R 5 then V has infinitely many bases. True. If V 1, V 2, V 3 is one basis, then V 1 +kV 2, V 2, V 3 is another basis for each integer k.

32. 32. Matrix I n is similar to 2 I n. False. I n is similar to only itself.

33. 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. 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. 35. If V and W are subspaces of R n, then their union VuW must be a subspace of R n 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. 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 R 4, then vectors V and W must be equal. True. Since A(V-W) = 0, V-W = 0 and so V=W.

37. 37. If V 1, V 2, …, V n and W 1, W 2, …, W n are two bases of R n, then there is a linear transformation T from R n to R n such that T(V 1 ) = W 1, T(V 2 ) = W 2, …, T(V n ) = W n. True. You can map a basis anywhere.

38. 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] | A 4 = I and B 4 =/= I. They cannot be similar.

39. 39. R 2 is a subspace of R 3. False. There are subspaces of R 3 of dimension 2, but the vectors in them are all three tuples, not 2 tuples.

40. 40. If an nxn matrix A is similar to matrix B, then A + 7I n must be similar to B + 7 I n. True. If P -1 AP = B then P -1 (A+7I n )P = P -1 AP + 7 P -1 I n P = B + 7 I n

41. 41. There is a 2x2 matrix A such that im(A) = ker(A). True. | 0 1 | is one such matrix. | 0 0 |

42. 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. 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. 44. If A 2 = 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. 45. For every subspace V of R 3 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. 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. 47. If the 2x2 matrix R represents the reflection across a line in R 2, then R must be similar to the matrix | 0 1 |. | 1 0 | True. Use the basis | / _____|/_______ |\

48. 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 -1 S) -1 A (A -1 S) will also work.

49. 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. 50. If A is any nxn matrix such that A 2 = 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 = A 2 V = AV.

51. 51. There is a 2x2 matrix A such that A 2 =/= 0 and A 3 = 0. False. If A 2 V =/= 0, and A 3 V = 0, then V, AV, A 2 V must be linearly independent. This is impossible in R 2.