Solutions to HW
Final Exam: Friday, May 11, 2012, 8:30-11:30 a.m. The exam will be cumulative, but with some emphasis on the material after the last midterm (HWs 23-32).
You should also be able to prove these theorems and extra problems.
Here are the 2011 final exam and the 2007 final exam (ignore problem 4).
Office hours during finals week: R 5/10, 4:00-5:15.
HW #32, due M 4/23: (this is our last HW assignment :) Read Sec 7.3 p.580-583 and Examples 7.25-7.28. Do: Sec 7.3: 1, 7, 15.
HW #31, due W 4/18: Read Sec 2.4 Examples 2.33 and 2.34. Do: Sec 2.4: 23, 25, 26. Hint for 25: The equation Ax=b has a solution iff b is in col(A); if A is symmetric, then col(A)=row(rref(A)) (why?).
Just for fun: the game of Lights On. You can play the book's version of the game (just one column of lights, as in class) by restricting attention to only the first column of the 5x5 array in the "full version" of the game. A solution to the 5x5 game is posted on the HW Solutions page.
HW #30, due M 4/16: Read p.442-444. Do: Sec 6.1: 22, 23, 26, 29, 30, 59, 63, 64. Also read p.50-51; and do: Sec 1.4: 27, 29, 31.
HW #29, due F 4/13: Read Sec 6.1 up to Subspaces (p. 437); may skip Examples 6.7 & 6.8. Do: Sec 6.1: 1, 3, 5-9, 10, 61, 62.
HW #28, due W 4/11: Read Sec 5.3 up to (but not including) QR Factorization. Do: Sec 5.3: 5, 7, 9, 10, 11, 12.
HW #27, due M 4/9: Read the rest of Sec 5.2. Do: Sec 5.2: 21, 24, 26. Also do these extra problems.
HW #26, due F 4/6: Read Sec 5.2 up to Orthogonal Projections. Do: Sec 5.2: 1, 3, 5, 23. Also do these extra problems.
HW #25, due W 4/4: Read Sec 5.1 up to but not including Orthogonal Matrices. Proof of Theorem 5.1 may be on exams. Do: Sec 5.1: 1, 3, 7, 11, 12. Also do these additional problems.
HW #24, due M 4/2: Read Sec 4.4: p. 298-300 and Example 4.29. Proof of Theorem 4.22abde may be on exams. Do: Sec 4.4: 1, 3, 17, 18, 30, 31, 33, 34, 40, 41, 42.
Midterm #2: Friday 3/30. Covers HWs 12-22 and their corresponding sections. Most or all problems on the exam will be similar to homework problems. You may also be asked to give definitions; and you should know the proofs of Theorems 3.4(d), 3.6, 3.9(a-d), 3.19, 4.3a,c,d. Solutions
HW #23, due W 3/28: Read Sec 4.3. Do: 7, 13, 15, 16. Also do these problems: (i) Let A be an n x n matrix, E an eigenspace of A. Prove that E is a subspace of R^n by showing that it satisfies the definition of subspace. (Hint: start by writing down the definition of eignespace.) (ii) Prove or disprove: if A is singular, B is non-singular, then AB is singular. (Recall that "singular" means "not invertible".)
HW #22, due M 3/26: Read Sec 4.2 up to Cramer's Rule. Also read about the Cross Product on p. 283. Optional: Area and Volume (p. 284) Do: Sec 4.2: 44, 49, 50, 51, 53, 54. Page 284: 2, 3a-e,f. Also do these problems:
(i) Prove that if B is obtained from A by one elementary row operation, then det A = 0 iff det B = 0; you may use Theorem 4.3 for this. (ii) Prove det A = 0 iff det rref(A) = 0; you may not use FTIM for this. (iii) Use the above to show A is invertible iff det A is nonzero.
HW #21, due F 3/23: Read Sec 4.2 up to Determinants of Elementary Matrices. You should learn the proof of Theorem 4.3a,c,d. Do: Sec 4.2: 3, 13, 15, 17, 19, 20, 39, 40. Typo in #20: should say "definition (3)".
HW #20, due W 3/21: Read Sec 4.1 (skip the last two examples). Do: 3, 7, 13, 15, 17-19, 21, 22, 23.
HW #19, due M 3/19: Read Sec 3.5 p.199-205. Do: Sec 3.5: 38, 39, 41, 47, 55, 56. Hint for 56: Use FTIM
HW #18, due F 3/9: Read Sec 3.5 Examples 3.45-3.47. Do: Sec 3.5: 17, 18, 19, 21, 22, 25, 40, 48. In 18-23 ignore null(A). For #21 and 22, read the middle paragraph of p.198. Hint for #48: Add the vectors! (Note: the answer in the back of the book to #48 is incorrect.) In all problems you may use a calculator or computer for tedious computations to find rref.
HW #17, due W 3/7: Read Sec 3.5 up to and including Example 3.44. May skip those parts of Example 3.41 that deal with the vector w. Proof of Theorem 3.19 may be on exams. Review Problem 21 of Section 2.3 (problem from HW#10). It's important that you understand and remember it well. Do: Sec 3.5: 11 (only for col(A)), 13, 33, 34, 39, 45, 49.
HW #16, due M 3/5: Read Sec 3.5 up to and including Example 3.40. Do: Sec 3.5: 1, 2, 3, 5, 6, 7, 8, 9, 10.
Note: In the solutions to Midterm #1 posted below, I've indicated which homework problem each problem on the exam was based on. You should look at whether you did that hw problem correctly when you turned it in, and compare your work with how you did on the exam. This should help you study better in the future.
I was curious how the pdf file format "works". While reading about it on wikipedia, I came across this:
"PDF graphics use a device independent Cartesian coordinate system to describe the surface of a page. A PDF page description can use a matrix to scale, rotate, or skew graphical elements."
HW #15, due F 3/2: Read Sec 3.3: Elementary Matrices, and The Fundamental Theorem of Invertible Matrices. Example 3.28 will help with problems 35-38 of the homework. Do: Sec 3.3: 25, 27, 35, 36, 37, 38, 39, 45, 46.
HW #14, due W 2/29: Read Sec 3.3 up to Elementary Matrices, plus The Gauss-Jordan Method for Computing the Inverse. Proofs of Theorems 3.6 and 3.9(a-d) may be on exams. Do: Sec 3.3: 14,15, 16, 21, 22, 23, 43, 49, 59.
Midterm #1: Friday Feb 24. The exam will cover HWs 1-11 and their corresponding sections. Most or all problems on the exam will be similar to homework problems. You may also be asked to give definitions; and you should know the proof of Theorem 2.6. Solutions
HW #13, due M 2/27: Read Sec 3.2, but skip Examples 3.16-3.18. Proof of Theorem 3.4(d) may be on future exams. Sec 3.3 up to Properties of Invertible Matrices. Sec 3.2: 30, 34, 35, 36, 44. Sec 3.3: 3, 11, 19.
HW #12, due W 2/22: Read Sec 3.1: p. 143-150; no need to memorize what "outer product" is (p. 145). Do: Sec 3.1: 13, 17, 23, 24, 29, 30, 31, 39c, 41.
HW #11, due F 2/17: Read Sec 3.1: up to Partitioned Matrices. Do: Sec 3.1: 1, 3, 5, 11, 18, 19, 20, 21. Sec 2.3: 33, 42-45.
HW #10, due W 2/15: Read Sec 2.3: Linear Independence. Do: Sec 2.3: 11, 20, 21, 25, 27, 28. In #20, for the definition of subset (the "horseshoe-like" symbol), see Appendix A (p. 634-635) or see Wikipedia.
HW #9, due M 2/13: Read Sec 2.3, up to Linear Independence (p.90-94). Do: Sec 2.3: 1, 3, 8, 9, 10, 13, 15, 18, 19. May use calculator or computer for tedious computations in finding rref: http://faculty.oxy.edu/rnaimi/home/onlineTools.htm.
HW #8, due F 2/10: Read Sec 2.2: Rank and Homogeneous Systems; skip "Linear Systems over Zp." Do: Sec 2.2: 11, 23(1,3,5,7), 35, 37, 41, 47, 49.
HW #7, due W 2/8: Read Sec 2.2 up to Homogeneous Systems; but ignore "Rank" for now. Do: Sec 2.2: 1-9(odds), 16, 19, 21, 25, 27.
HW #6, due M 2/6: Read Sec 2.1. Do: Sec 2.1: 1, 3, 11, 13, 15, 17, 23, 28, 29, 32, 34, 35.
HW #5, due F 2/3: Read the rest of Sec 1.3. Do: Sec 1.3: 7, 13, 18abcd , 19, 29, 33. (Hint for #33: see Example 1.26.)
HW #4, due W 2/1: Read Sec 1.3 up to "Planes in R^3", plus Example 1.25. It'll help you if you also preview the rest of Sec 1.3 before class. Sec 1.3: 1, 5, 11, 15, 16, 23, 28.
HW #3, due M 1/30: Read Sec 1.2, Projections (p. 24-25). Do: Sec 1.2: 31, 41, 54, 62-64. For 41 see Figure 1.36. Try to do 64 without reading the hint below. Hint for 64a: First explain why proj_u(v) is a scalar multiple of v. Then prove that if c is any scalar, proj_u(cu) = cu. Hint for 64b: Use 63.
HW #2, due F 1/27: Read Sec 1.2 up to projections; may skip the proof the Theorem 1.5. Pay attention to the remarks on page 16. Also read page xxiii (before Section 1); it has some really good advice. Do: Sec 1.2: 5,11,17, 25, 44, 47, 48, 52.
HW #1, due W 1/25: Read Sec 1.1. Advice: Read the entire section, not just what seems necessary for the HW problems; otherwise you may miss the "big picture." Do: Sec 1.1: 1d, 2d, 3c, 4c, 5a, 6, 9, 15, 17, 20, 23,24.
You should turn in all problems, but the boldface ones are the ones that'll likely get graded.