1. Math 307 Spring, 2003 Hentzel Time: 1:10-2:00 MWF Room: 1324 Howe Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://www.math.iastate.edu/hentzel/class.307.ICN Text: Linear Algebra With Applications, Second Edition Otto Bretscher

2. Previous Assignment Friday, Mar 28 Chapter 5.5 Page 240 Problems 1 through 43 1. If matrix A is orthogonal, then matrix A 2 must be orthogonal as well. True. Orthogonal matrices are closed under multiplication.

3. 2. The equation (AB) T = A T B T holds for all nxn matrices A,B. False. The correct version is (AB) T = B T A T. | | 0 1 | | 0 0 | | T = | 1 0 | T = | 1 0 | | | 0 0 | | 1 0 | | | 0 0 | | 0 0 | | 0 1 | T | 0 0 | T = | 0 0 | | 0 1 | = | 0 0 | | 0 0 | | 1 0 | | 1 0 | | 0 0 | | 0 1 |

4. 3. If A and B are symmetric nxn matrices, then A+B must be symmetric as well. True. Symmetric matrices are a subspace.

5. 4. If matrices A and S are orthogonal, then S -1 A S is orthogonal as well. True. The product of orthogonal matrices is orthogonal.

6. 5. All nonzero symmetric matrices are invertible. False. A counter example is | 1 1 | | 1 1 | which has rank 1.

7. 6. If A is an nxn matrix such that A A T = I, then A must be an orthogonal matrix. True. Since A is square, A T = A -1 and so A T A = I. Thus, the columns of A are orthonormal.

8. 7. If V is a unit vector in R n, and L = span[V], then Proj L(X) = (X o V)X for all vectors x in R n. False. This does not project into. L(X) = (XoV) V

9. 8. If A is a symmetric matrix, then 7 A must be symmetric as well. True. The symmetric matrices are a subspace.

10. 9. If T is a linear transformation from R n to R n such that T(E 1 ),T(E 2 ),..., T(E n ) are all unit vectors, then T must be an orthogonal transformation. False. | 1 1 | is a counter example. | 0 0 |

11. 10. If A is an invertible matrix, then the equation (A T ) -1 = (A -1 ) T must hold. True. I = (A A -1 ) T = (A -1 ) T A T

12. 11. If A and B are symmetric n x n matrices, then A B B A must be symmetric as well. True (A B B A) T = A T B T B T A T = A B B A

13. 12. If matrices A and B commute, then matrices A T and B T must commute as well. True. A T B T = (B A) T = (A B) T = B T A T

14. 13. There is a subspace V of R 5 such that dim(V) = dim(V _|_ ), where V _|_ denotes the orthogonal complement of V. False: Dim (V) + Dim(V _|_ ) = 5 so they cannot be equal.

15. 14. Every invertible matrix A can be expressed as the product of an orthogonal matrix and an upper triangular matrix. True. This is the A = Q R decomposition.

16. 15. If X and Y are two vectors in R n, then the equation |X+Y| 2 = |X| 2 + |Y| 2 must hold. False. This only holds when the vectors are orthogonal.

17. 16. If A is an n x n matrix such that | A U | = 1 for all unit vectors U, then A must be an orthogonal matrix. True. This means that | A X | = | X | for all X and so A is an orthogonal matrix,

18. 17. If matrix A is orthogonal, then A T must be orthogonal as well. True. If A is orthogonal, then A must be square and A T A = I means that A A T = I so A T is orthogonal as well.

19. 18. If A and B are symmetric n x n matrices, then AB must be symmetric as well. FALSE. | 0 1 | | 1 0 | = | 0 0 | | 1 0 | | 0 0 | | 1 0 | is a counter example

20. 19. If V is a subspace of R n and X is a vector in R n, then the inequality X o Proj V X >= 0 must hold. True. Suppose Y+N = X where Y is in V and N is perpendicular to V. Y = Proj V X and Y o Y >= 0 and Y o N = 0 Y o Y + Y o N = Y o X So YoX = YoY >= 0.

21. 20. If A is any matrix with ker(A) = {0}, then the matrix A A T represents the orthogonal projection onto the image of A. False. This is true if the columns of A are orthonormal. If not, one has to use A (A T A) -1 A T

22. 21. The entries of an orthogonal matrix are all less than or equal to 1. True. Since there squares all add to 1, each has to be at most 1.

23. 22. For every nonzero subspace of R n there is an orthonormal basis. True. This is the Gram-Schmidt Process.

24. 23. | 3 -4 | is an orthogonal matrix. | 4 3 | False. The columns are not of unit length.

25. 24. If V is a subspace of R n and X is a vector in R n, then vector proj V X must be orthogonal to vector X- Proj V X True. The projection is perpendicular to the space and proj V X is in the space, so proj V X is perpendicular to X-proj V X

26. 25. If A and B are orthogonal 2x2 matrices, then A B = B A. False. | 1 -1 | | 1 1 | = | 0 1 | | 1 1 | | 1 -1 | | 1 0 | ----------- --------- Sqrt[2] Sqrt[2] | 1 1 | | 1 -1 | = | 1 0 | | 1 -1 | | 1 1 | | 0 -1 | ----------- --------- Sqrt[2] Sqrt[2]

27. 26. If A is a symmetric matrix, vector V is in the image of A and W is in the kernel of A, then the equation VoW = 0 must hold. True. VoW = V T W = (AX) T W = X T A T W = X T A W = 0

28. 27. The formula ker(A) = ker(A T A) holds for all matrices A. True. If AX = 0, then A T (AX) = 0. If A T (AX) = 0, then X T A T AX = 0 so (A X)o(AX) = 0 and thus AX = 0.

29. 28. If A T A = A A T for an n x n matrix A, then A must be orthogonal. False. It is true for any symmetric matrix including | 1 1 |. | 1 1 |

30. 29. If A is any symmetric 2x2 matrix, then there must be a real number x such that X-x I 2 fails to be invertible. True. det | a-x b | = (a-x) 2 – b 2 = | b a-x | (a+b-x)(a-b-x) so if x = a+b or x = a-b, the matrix will not be invertible.

31. 30. If A is any matrix, then matrix 1/2(A-A T ) is skew-symmetric. False: If A is not Square, then A-A T is not defined. If A is Square, then it is true. ( 1/2(A-A T )) T = 1/2 (A T – A ) = -1/2 (A - A T ).

32. 31. If A is an invertible matrix such that A -1 = A, then A must be orthogonal. False. A = | 1 b | Squares to the identity. | 0 -1 |

33. 32. If the entries of two vectors V and W in R n are all positive, then V and W must enclose an acute angle. True. Since V o W is positive, Cos[theta] is positive and theta < Pi/2.

34. 33. The formula (ker B) _|_ = im( B T ) holds for all matrices A. True. It simply says that B ker(B) = 0.

35. 34. The matrix A T A is symmetric for all matrices A. True. (A T A) T = A T A.

36. 35. If matrix A is similar to B and A is orthogonal, them B must be orthogonal as well. False. | 1 -1 | | 1 -1 | | 1 1 | | 0 1 | | 1 1 | | 0 1 | -------- Sqrt[2] 1/Sqrt[2] | 0 -2 | | 1 1 | | 1 1 | | 0 1 | 1/Sqrt[2] | 0 -2 | | 1 2 |

37. 36. The formula Im(B) = Im(B T B) holds for all square matrices B. False. | 0 1 | has image | x | | 0 0 | | 0 | B T B = | 0 0 | | 0 1 | = | 0 0 | has image | 0 | |1 0 | | 0 0 | | 0 1 | | x |

38. 37. If matrix A is symmetric and matrix S is orthogonal, then matrix S -1 A S must be symmetric. True. S T A S is symmetric when A is.

39. ` 39. There are orthogonal 2x2 matrices A and B such that A+B is orthogonal as well. True. | ½ -Sqrt[3]/2 | | ½ +Sqrt[3]/2 | | +Sqrt[3]/2 ½ | | -Sqrt[3]/2 ½ | Are two orthogonal matrices which add to I 2.

40. 40. If | AX | <= | X | for all X in R n, then A must represent the orthogonal projection onto a subspace V of R n. False: Let A = ½ I.

41. 41. Any Square matrix can be written as the sum of a symmetric and a skew-symmetric matrix. True A = ½ (A +A T ) + ½ (A – A T ).

42. 42. If x 1, x 2, …, x n are any real numbers, then the inequality n n (SUM x k ) 2 <= n SUM (x k 2 ) must hold. k=1 k=1 True | AoB| 2 <= |A| 2 |B| 2 and use A = the vector all of whose entries are 1.

43. 43. If A A T = A 2 for a 2x2 matrix A, then A must be symmetric. True. A(A T – A) = 0 If A is not symmetric, then the first and second columns of A have to be zero.