MATH 700 -- Linear Algebra
Professor Matt Miller (miller@math.sc.edu)
Section 1, TTh 4:45-6:00 + epsilon, LC 310

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Text: Linear Algebra Done Right by Sheldon Axler, published by Springer, 2nd. ed., 1997.

Class topics and problems Assignments are made on a weekly or week and a half basis if there is a day off for some reason or another. Try to keep up and even push ahead if you can.
  • Aug. 21-28. Vector spaces, subspaces, fundamental examples, sums and direct sums. Read the preface to the text and chapters 1 and 2. You will observe that I assign homework BEFORE we talk about a section. That is because I really want you to READ the section, and struggle a bit; then the class and my lecture will make more sense to you, and you will be in a better position to ask good questions. Attempting problems and seeing where you get stuck is much more useful (albeit painful) than churning through problems that you basically already know how to do! In fact, you will find that the text itself is written in this style: many important results are developed in the exercises, and small steps are often left to you.
    ---Problems marked with a * will be collected one week after they are assigned. From Chapter 1 do problems 4-8, 9*, 10, 13*, 15* (due 8/28). From Chapter 2 do 1, 3-6, 7*, 8, 11*, 14, 16, 17*, QA06 #7, QJ07 #2*, QA07 #1, QJ08 #1 (due 9/9). Feel free to be sketchy on many of the problems, as there are a lot of them, but write up the starred ones carefully as I described on the information sheet.
  • Sept. 2-4. Independence, spanning, and bases. Continue to read chapter 2 and to work on the problems given above. Problem A from class will be due on 9/11. It is quite important that you be familiar with Gauss-Jordan elimination as a method for solving systems of linear equations. Find an undergraduate linear algebra text and review this material (I understand that Wiki has it too).
  • Sept. 9-11. Bases (continued) and the maximum principle, linear transformations. Do the problems from the file that has been emailed to you; the * problems from pages 23-42 will be due on 9/11 (don't forget problem A*), and the ones from pages 43-62 will be due on 9/16. NOTE: when a problem asks you to find a basis, what this really means is FIND a basis AND PROVE that it is a basis. When you are asked to determine a dimension, there may occasionally be some external information that gives this to you, but in most cases, you will need to determine a basis. Review Quiz B will be on Sept. 11. Starred problems C(c)* and E* from Handout 1 (the green sheet) will be due on 9/18. Read chapter 3 of the text.
  • Sept. 16-19. Linear transformations (cont.), matrices, dual spaces and maps. Read chapter 3 of the text. Do exercises 1, 2, 3*, 4, 5*, 7, 8, 12*, 14, 15, 19, 20*, 21, 22*, 23, 24. Starred problems will be due on 9/23. After you have done the problems on the construction of V/W, do the following problem B*: determine a linear map pi:V-->V/W with the property that N(pi) = W and pi is onto. Show that there is a map s: V/W-->V so that pi composed with s is the identity on V/W, and show that V is the direct sum of N(pi) and R(s). Also do the following problem F*: Give an example of (an infinite dimensional) vector space V over the reals and a linear transformation T:V-->V that is 1-1 but not onto. Give a transformation S:V-->V that is onto but not 1-1. What do these results say about Theorem 3.21 in the text? Problems B* and F* will be due on 9/30, and are printed up nicely in Handout 2. (typos corrected on 10/14)
    Set aside the hours of Dr. Schep's class TTh 2:00-3:15 (9/16, 9/18, 9/23, 9/25) for the next two weeks for optional problem sessions. I hope you will do most of the talking! Also reserve the Friday 3:30-4:45 slot on Sept. 19th for a make-up class (I will be out of town on October 16).
  • Sept. 23-26. Dual spaces and maps (cont.), quotient spaces, Test #1. This will give us a first look at the concepts of canonical map, natural map, and universal mapping properties. This can seem intimidating at first, but once mastered, these are powerful tools. See below about the exam. We also have just a bit of tidying up to do with matrix representations of a linear transformation: How does the matrix for T: V-->W change when we change the bases of V and W? Apparently Axler found this matter a tad unpleasant, so he deferred the discussion to pp. 214-216. Read these and do problems 1, 2*, 3 and 4 on page 244 for October 2. Also show that similarity of square matrices is an equivalence relation (consider this a * problem). We had a short Quiz #1 to warm up to the exam.
  • Sept. 30-Oct. 7. Change of basis, invariant subspaces, eigenvalues and eigenvectors. The goal is to begin to understand the structure of operators on finite dimensional complex vector spaces, especially the existence and information obtainable from upper triangular form. Note that there is no class on Oct. 9 (Fall Break). Read chapter 5 of Axler. Prop. 5.21 is very important. Do problems 1, 2, 3, 4*, 5, 8, 10*, 11*, 12, 14*, 17-21 for Tuesday, October 14.
  • Oct. 14. Projections and structure of real and complex operators. I hope we will also find time to talk about complexification of real operators. Finish up reading chapter 5 and go on to chapter 6, which we will begin on October 21. Do 20*, 21*, 22 from chapter 5, QJa00 #1, and problems 5, 10*, and 12* from chapter 8 (you don't need anything from chapter 8 to do these--all can be easily done with the information that you already have; you just need to know that an operator T is nilpotent if T^r = 0 for some natural number r) for Tuesday, October 21. Also work on the problems from Handout 3, which will be due on Tuesday, October 28 instead of the optimistic Thursday, October 23. You guys are killing me with the homework!!
  • Oct. 21-23. Complexification (cont.), inner produuct spaces. There will be a Quiz #2 for sure. For Thursday, October 30, do Axler, chapter 6 #2, 9, 10*, 11, 13*, 15, 16, 17*, 18, 19, 20*, 27, and chapter 8, #3. Don't get confused by our notation for lists and for inner products; we will also be using T* now for the adjoint of T, which is related to, but not exactly the same as T*, the dual map that we discussed earlier--but don't worry, the context will always make it clear what we are talking about!
  • Oct. 28-30. Inner product spaces (conclusion), self-adjoint operators. There is no class on Nov. 4 (Election Day). For Thursday, November 6, do Axler, chapter 6 #28*, 29-31, the exercises given in class, and Chapter 7 #1, 2, 3, 4*. For Thursday, Nov. 13, do Chapter 7 #6*, 8, 9*, 10, 11, 15*, 16.
  • Nov. 6-13. Self-adjoint and normal operators, the Spectral Theorem(s), generalized eigenvectors. Read Prop. 7.18 and its proof, pages 140 to the end of the first full paragraph on p. 143. Read about isometries: pages 147 to the bottom of p. 150 (This material will NOT be on Exam #2). Read Chapter 8, pages 163-176, 179-187. For Thursday Nov. 20, do Chapter 8 problems #1-4, 6*, 7*, 8, 10, 11*.
  • Nov. 18-25. Structure of nilpotent and general complex operators, Cayley-Hamilton Theorem and Jordan canonical form, Exam #2. For Thursday, December 4, do Chapter 8 problems #12, 14*, 15, 16, 17*, 22*, 23-27, 28*. When Axler asks for "an example", instead give ALL examples, up to similarity (rearrangement of blocks).
  • Dec. 2-4. Determinants and the traditional approach to the characteristic and minimal polynomials. We use the traditional definition of the char. poly. and complexification to prove the C-H Theorem over the reals, and we state it for all fields; we extend the minimal polynomial characterization of diagonalizable operators to fields in general (you did this in the homework over the complex numbers).

    • Exams
  • FIRST EXAM: Friday, September 26 at 3:30 (class #11.5). it seems most people are happy pushing the exam back by one day, so there is more time to absorb the material; if this causes a time conflict for you, contact me as soon as possible. The exam will cover through chapter 3, plus the material on dual and quotient spaces (and their maps) as covered in class through Tuesday, September 23. Do study ALL the homework problems, including but not limited to, the starred ones. The "drop deadline" is Thursday, October 2, so if you have concerns about continuing in the course, you should have a reasonable idea how things are going by then.
  • SECOND EXAM: Friday, November 21 (class #25a), 3:30-5:00, LC 101. This exam will cover the material from Chapters 5, 6, and 7 that we discussed in class, and a bit of 8. You will be expected to know the results of chapter 4, but this is material that you should have seen already (and will see again in MATH 701).
  • FINAL EXAM: Thursday, December 11 (5:30-8:30 pm or so). There will be a review session before the exam, on Wednesday at 11:00 am. You will be allowed to bring one standard sized sheet of paper, with writing only on one side, which you will turn in with your exam; it may contain any material you wish. Even if you feel you do not need such a sheet, preparing one can be a good way to organize your thoughts, determine what is of primary and secondary importance, etc. On the final you may use determinant arguments and the statements of results that we stated in the last week, even if we didn't prove them in class. But there will be NO problems for which a determinant argument is actually necessary. While the final exam will be comprehensive, and the percent score will replace the lower of your two exam scores if this helps you, the content will lean more heavily on the material of Sept. 30 onwards (test #2 and afterwards). As a further incentive to do well on the final I will use 0.8 times your percent score to replace your other exam score, but again, only if this helps you. So for example if you get 125/150 on the final, an 83 will replace your lowest score, and 67 to replace your second lowest score. Quiz points will be tossed into the total as bonus points.


    • Last modified: December 8, 2008