MATH 700 -- Linear Algebra
Professor Matt Miller (miller@math.sc.edu)
Section 1, MW 4:40-5:55, LC 303B

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Text: Linear Algebra by S. H. Friedberg, A. J. Insel, and L. E. Spence, published by Prentice Hall, 4th ed., 2003, ISBN 0-13-008451-4.

Class topics and problems
  • Aug. 22-28. The basic language: vector spaces, subspaces, independence, dependence. Read the preface to the text and sections 1.1-1.6, 3.1 of the text. 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.
    ---Problems marked with a * will be collected on Monday, August 29. Do a sampling of problems 1-3 from 1.1, 1-3 from 3.1, and problems 1, 11, 12, 14*, 15, 18, 20, 21 from 1.2, problems 1, 3, 4, 5, 6*, 12, 15, 17, 18, 19*, 20*, 21, 23*, 24, 26, 28*, 30*, 31abc*d from 1.3. Also do problems Aabc*, B, and C* from the handout. 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.
    ---Problems marked with * will be collected on Wednesday, Aug. 31. In 1.4 do 1, a selection from 3, a selection from 5, 6*, 10, 11, 13*, 14*, 15, 16, and in section 1.5 do 1, a selection from 2, 6, 7, 9* 10, 12, 13a* (not of characteristic two means that there is an element 1/2 in F), 20.
  • Aug. 29-Sept. 6. Bases and the maximum principle. Read sections 1.6, 1.7, and 3.4 of the text. It is quite important that you be familiar with Gauss-Jordan elimination as a method for solving systems of linear equations. Do enough exercises in 3.4 that this method becomes familiar to you. For section 1.6 do problems 1, 2 (sampling), 3 (sampling), 5, 7, 8, 13, 14, 15*, 20*, 24*, 26*, 29*, 33, 34, 35. The ones that are starred will be collected on Wednesday, 7 September (Monday is a holiday). Do problems 1, 2*, and 4 from section 1.7 for Monday, 12 September; there will be additional problems from section 2.1. 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.
  • Sept. 7-11. Bases and the maximum principle (conclusion). Read ahead in section 2.1 and do problems 1, 2 5, 6, 7, 8, 10, 11, 12, 13*, 14c*, 15, 17*, 21*, 26a*b, 31, 32, 40 for Monday, 19 September.
  • Sept. 12-18. Linear transformations and matrices. Do the work from section 2.1 as described above; then proceed to sections 2.2 and 2.3. Despite the overpowering notation these are basically telling you nothing more than you know a lot about linear transformations (on finite dimensional vector spaces) already because you know about their representations as matrices. For Wednesday, Sept. 21, do problems 1, a sampling of 2, 3, 4, 5ac*d, 6, 7, 8*, 9*, 10, 11*, 13, 15* from 2.2. Please note that I will be out of town Friday 16 September-Sunday 18 September.
  • Sept. 19-25. Matrices (cont.), isomorphisms, and change of coordinates. Finish up reading 2.3, and go on to 2.4 and 2.5. You'll note that I stated Theorem 2.20 on page 103 already (see class notes of 14 Sept.); the difficulties of these sections are really more notational than conceptual. The only real big result is Theorem 2.23 on page 112. Do problems 1, 3a, 4acd, 5, 6, 7, 9*, 11, 12, 13*, 15, 16* (Hint: how does R(T^k) compare with R(T^(k+1) and how does N(T^k) compare with N(T^(k+1))?), 17, 18* from 2.3, to be turned in on Monday, 26 Sept.
  • Sept. 26-Oct. 2. Dual spaces and exam 1. For Monday, 3 Oct., do problems 1, 2 (sampling), 3 (sampling), 4*, 5, 6, 9*, 10*, 15, 16, 17, 20, 23*, 24 from section 2.4. For Monday, 10 October do problems 1, 2 (sampling) 4, 6 (sampling), 7*, 9*, 10 from 2.5; and do problems 1, 2 (sampling), 3*, 8, 11, 13*, 14, 15*, 19* from 2.6. The exam on Wednesday, 28 Sept. will cover through section 2.4 only.
  • Oct. 3-9. Dual spaces (cont.), linear systems and ranks Note that the homework due date for section 2.6 has been pushed back to Monday, 10 Oct. Read all the text of chapter three, just leaving out the applications. Try to fill in the proofs where they have been omitted. Our main results will be that rank of a transformation is equal to the column rank of a matrix representative, which in turn is computed by the row rank of this matrix (so all three are equal, and complementary to the nullity). We will state the most important results about the reduced row echelon form. However most of this chapter is trivial from the point of view of linear transformations, involving as it does, mostly how to do matrix computations and how to interpret the results. Homework will be due on Wednesday, 12 Oct. Section 3.1: 1, 3, 5, 7, 8; section 3.2: 1, 2e, 2f, 6c, 6e, 14, 19*, 21, 22; section 3.3: 1, 2 (sample), 7 (sample), 9, 10*; section 3.4: 1, 2 (sample), 3, 4 (sample), 5, 8, 10*.
  • Oct. 10-16. Determinants We are going to race through this chapter 4; I am hoping that you have seen much of it already. In section 1, do 1, a sampling from 3 and 4, 5-10, 11* (suggestion: figure out how delta behaves with respect to row operations, and think about what the possible reduced row echelon forms of a 2x2 matrix can be); in section 2, do 1, a sampling from 5-7, a sampling from 13-22, 25, 26, 28, 29; in section 3 do 1, 10, 11, 12*, 13, 15*, 16, 18*, 21, 22abc*, 23*; and from section 4, do 1, 3b, 3d. Finally in section 5, do 1, 13, 16, 19, 20. These problems will be due on Monday, October 24. Please note that I will be out of town Friday 14 October-Sunday 16 October.
  • October 17-23. Diagonalization We are going to slow down just a bit now, and skip 5.3; but in truth 5.4 is the only difficult section. Anyhow for Wednesday, Oct. 26, do section 1, problems 1, a selection from 2 and 3, 5, 7, 8a*b, 9, 11a*b*c, 12, 14*, 15, 19, 20, 21, 22, 23*; then do section 2, do 1, a sampling from 2, 7, 9, 10, 11, 12*,18a*b, 20, 22. Please note that I will be out of town Thursday, 20 October-Sunday 22 October.
  • Oct. 24-30. Diagonalization (cont.), invariant subspaces, and the Cayley-Hamilton Theorem Do problem A* for Monday, 31 October. Section 5.4 is deep and important; you will want to read it very carefully. To get started do problems 1, sampling of 2, 4, 6b, 7, 9b, 10b, 11, 12, 13, 15*, 16, 17, 18*, 19, 20* for Wednesday, 2 November. There will be more problems from this section. Please note that I will be out of town Thursday, 27 October-Sunday 30 October.
  • Oct. 31-Nov. 6. Invariant subspaces (cont.) and the minimal polynomial. Clean up the minus signs in my computation of the characteristic polynomial of a companion matrix; sorry about that. Finish off problems 21*, 22*, 23*, 24*, 36 in 5.4, read section 7.3 and do problems 1, a sampling from 2-4, 5, 6, 7, 8*, 9*, 10, and 12. You may use techniques from 7.3 to do problems in 5.4 if you wish. These will be due on Monday, 7 November. Please note that I will be out of town Friday, 4 November-Sunday 6 November.
  • Nov. 7-13. Inner product spaces, orthonormal bases. Read sections 6.1 through 6.3. In 6.1 do problems 1, 3, 5, 9*, 10*, 11, 12*, 14, 17, 19, 21, 22a*. In 6.2 do problems 5, 7, 8, 9, 11*, 15a*, 18*. Finally in 6.3 do 1, 2a, 3a, 6, 8, 9, 10*, 12a*, 18. The starred problems are due on Friday, November 18. There will be a problem session on Friday, November 11 at 4:30.
  • Nov. 14-20. The adjoint, Test #2. Read section 6.4.
  • Nov. 21-27. Recap of characteristic and minimal polynomials, the adjoint (cont.) Finish reading section 6.4, do problems 1, 3, 4*, 6, 7*, 8*, 12 (to turn in on Wednesday, Nov. 30), fill in the gaps in the proof of Theorem 6.15, and go on to section 6.5.
  • Nov. 28-Dec. 2. Normal and self adjoint operators, unitary and orthogonal operators, diagonalization and projection We will hit the highlights of sections 6.4 and 6.5 with some results just stated. We will also retrace our steps to Theorem 6.6 in section 6.2, and some of the results that depend upon it in subsequent sections. Time permitting, we will also look at the classical version of the Principal Axis Theorem. Our extra class is Friday, 3:45-5:00, concluding in time for the departmental festivities (which are mandatory!). Give problems 1, sample of 2, 3, 4, sample of 5, 7, 9, 10, 11, 12, 15, 17, 18, 19, and 21 from 6.5 a try. After doing Theorem 6.6, you should be able to do 6, 13, 14 and 16a from 6.2, and 9 from 6.4. Don't forget the Department Party at 5:00 on Friday, December 2!

    • Exams
  • FIRST EXAM: Wednesday, September 28 (class #11). The exam will cover through section 2.4; it will not include sections 2.5 and 2.6. Do study all the homework problems, including but not limited to, the starred ones. The "drop deadline" is Sept. 29, so if you have concerns about continuing in the course, let me know if you need your exam graded overnight (this will apply to almost no-one).
  • SECOND EXAM: Wednesday, November 16 (class #25) This exam will cover from 2.5 through chapters 3 and 4, chapter 5 (omitting 5.3), 7.3.
  • FINAL EXAM: Wednesday, December 7 (5:30-8:30 pm or so). There will be a review session before the exam, on Monday at 6:30. The content will be that of the text: chapters 1, 2, less intensively 3 and 4 (but material from these chapters is of course used throughout), 5 except for 5.3, 7.3, and 6.1 through 6.5. 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. I will be on campus through the weekend and next week; to see me just call or send an email.

    • Last modified: December 2, 2005