MATH 544 -- Linear Algebra
Professor Matt Miller (miller@math.sc.edu)
Section 501, MWF 10:10-11:00 in LC 310

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Text: Linear Algebra by David C. Lay, published by Addison-Wsley, 3rd ed. (regular or updated), 2003.
Office hours: MW 2-4 and TTh 1-3, no appointment needed, and of course by appointment. I will be in either LC 300i or LC 411.

My teaching style: 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.
Homework guidelines: Weeks are indicated on a Monday-Friday basis; don't expect to be able to do them all at once, but push ahead as much as you are able. There are many more exercises than you can do in a week. I leave it to your judgment how many repetitions of similar problems you need to do to be confident that you can do such problems on an exam. Problems marked with a * will be collected in class on Monday of the following week, unless otherwise indicated. You will note that the Practice Problems (henceforth PP) have full solutions after the Exercises, but don't peek until you have given them a fair try! Anything below the green line has not yet been updated for this class. I will try to keep this set of problmes small, so at the rate of one a day you should be able to complete the set comfortably.
Resources: I have a copy of the Student Study Guide and the Instructor's Solition Manual in my LC 300i office. Whenver the door is open you may feel free to come in and inspect these; you don't need to say anything more than "hi!", but of course I'll be happy to discuss the material.

Class topics and problems
  • Jan. 12-16. Solution of linear systems, terminolgy, row reduction, matrix form Ax=b. Read the Preface, Note to students, etc. and then sections 1.1-1.4 of the text. In 1.1, do PP #2-4 and exercises #3, 4, 7-9, 11-14, 18, 23, 24, 25*, 33, 34. In 1.2 do PP #1-2, and exercises 1, 2, 7-9, 11, 12, 15, 16, 21, 22, 23, 24, 29, 30*, 31. Section 1.3 will be mostly review, but the definition on page 35 is very important, and is probably new to you. Memorize it, and study associated discussion that takes place before and afterwards. Do PP #2 and exercises #22*, 23, 24, 25, 26a*, 26b. In section 1.4, Theorem 4 is very important--understand it! Do PP #1, 2, and exercises #1-12, 25-34. Starred problems will be due on Wed., Jan. 21.
  • Jan. 21-23. Geometry of homogeneous and non-homogeneous linear systems, uniqueness and spanning issues. Read sections 1.4 and 1.5. Pay particular attention to Theorem 3 in 1.4; this is the basis of many homework problems and ideas that we will need later on. In 1.4 do problems #1-12 as needed for practice, 17-20, 25, 26*, 27-31, 32*, 33, 34*. In section 1.5 do PP #1, 2, and exercises 1-12 as needed, 14, 15, 16*, 21, 23-25, 26*, 33-35, 39-40. Problems marked with a * are due on Monday, Jan. 26.
  • Jan. 26-30. Linear independence of vectors, linear transformations. Read sections 1.7, 1.8, and 1.9 (as you will see I am sort of blending 1.8 and 1.9). Section 1.7 is one of the most important in the course, and will require repeated reading. I am assigning many, many problems. Feel free to use your calculators or other software to do the row reductions. Do as many of these as you need to in order to master the ideas: 1-8, 15-20, 23-28, 33-40. Problems that I would like you to do for sure are #9, 10, 21, 22, 30*, 31, 32*, 36*, 38*, the Exercise* given in class, with the starred problems to be turned in on Monday, Feb. 2.
  • Feb. 2-6. Linear transformations and their geometry, 1-1, onto. From section 1.8 do the following; use your calculator to do the arithmetic on the matrices! PP #1-3, Exercises #1, 3-6, 9-12, 19-22, 24-27, 29-31, 34-36. Some of the tougher problems that might appear in some fashion (along with their odd-numbered neighbors) on the first exam are 12*, 20*, 24*, 26*, 34*. These will be due on Feb. 9 (if you turned some in on the 2nd, I'll return them, and you can work on them some more). I apologize for the confusion. To save space many authors write vectors as rows instead of columns. As long as the problem s entirely self-contained and doesn't need the correspondance between transformations and matrices, this is fine. But once matrices come into the picure we will almost always take vecors to be columns (the only exception I can think of will be the "row space" of a matrix). In section 1.9 do the PP and then #1-6, 8, 10-13, 14*, 15-18, 21, 23-25, 26*, 28-32, 36* (also due on 2/9/09). The matrix of a linear transformation will be on the first exam, but the geometry of transformations and the 1-1 and onto ideas will not be.
  • Feb. 9-13. Test #1, Difference equations, matrix algebra, inverses. Read section 1.10, pp. 97-99. Do problems #9, 11-13 and finish exploring the example we did in class. Do the same thing with #13, that is, determine what happens to the population total over time, and what the percentage profile looks like in each case. Section 2.1 is very long, but mostly very easy, and likely material that you have seen before for the most part. The main new idea is that matrix multiplication is actually a very natural operation if we consider that it represents composition of linear transformations. Notice that you often do not need to compute an inverse (or not go all the way) to determine if a matrix is (theoretically) invertible or not. Do problems 1-7, 10*, 11, 12*, 15, 16, 18, 19, 21, 23, 24*, 25, 27, 28, 29. Turn in the starred problems on Monday, Feb. 16.
  • Feb. 16-22. Inverses (continued), determinants. Read section 2.2, and do problems PP #1, 2, and exercises #1, 3, 5-10, 13, 14, 16*, 17, 18*, 21, 22*, 23, 24, 31, 32. Read section 2.3 and do problems #1-9 odd, 8, 11-13, 15-18, 20*, 22*, 27-33, 34* (Hint: use the matrix representations), 37, 40 (this last one is a bonus problem). Turn in the starred problems on Monday, Feb. 23. Coming attractions: read chapter 3, The multiplicative property of det,the connection with invertibility, and the behavior with rspect to row and column operations are important; Cramer's Rule is not, although the formula (4) on page 203 is useful from a theoretical point of view (awful from a computational point of view!).
  • Feb. 23-27. Determinants (continued). Please ignore exercises 15-18 on page 191--this is a deadly trap. Homework will be drawn from section 3.1 #1-4, 9, 10* (show the work), 19-22, 33-37, 38*, 41, 43, 44, 46, section 3.2 #1-7, 11, 15, 17, 18*, 23, 24*, 25, 27-33, 34*, 35, 36*, 37, 38, and section 3.3 #11, 12, 17-21, 23, 24*, 27, 28*, 31 (Bonus). Once you feel comfortable using row and column operations and cofactor expansion, feel free to use the det command on your calculator. The starred problems are due on Monday, 2 March.
  • Mar. 2-6. Vector spaces, Test #2 Read sections 4.1 and 4.2. In 4.1 do PP #1, 2, and Exercises #1-6, 8-11, 13, 14, 21-24, 25-30 if you plan on taking abstract algebra some day, 31-33. You will find that many of the ideas and computations in 4.2 are familiar; we are just using new words.
  • Mar.9-15. Spring Break
  • Mar. 16-20. Subspaces, transformations, column and null spaces, bases Read sections 4.1, 4.2, 4.3. You already have the list of problems for 4.1, just note that 8* and 32* are to be turned in on Wednesday, 25 March. For this same date, do 4.2 PP #1, 2, Ex #1-5, 6*, 7-9, 14, 15, 17-19, 23, 24*, 25, 26, 30, 31, 32*, 33, 35, 36*, and from 4.3 PP #1-3, Ex #1-5, 11, 13, 14, 19, 20*, 21, 22, 23, 24*, 31-33.
  • Mar. 23-27. Coordinates of vectors and transformations, dimension, bases Read sections 4.4, 4.5, 4.6 and pages 329 (middle)-330. There we read about linear operators on a vector space V and how to represent these by a matrix, as discussed in class. This will replace the discussion of the change of coordinates matrix (and we will also omit section 4.7). For Monday, 30 March, do 4.4 PP #1, 2, Ex #1, 3, 5, 6*, 7, 8, 13-19, 23-27, 32*. Also do 5.4 #11, 12*, 17b, 29-31. For these last problems the key is to solve systems of the form Ab_1 = x_1b_1+x_2b_2+x_3b_3, etc. where _ denotes subscript, or equivalently Bx = Ab_1, etc., where B is built up from the three b_i column vectors and x is the usual column vector of unknown x_i's. Use your calculator!
  • Mar. 30-April 5. Bases, rank plus nullity theorem, Markov chains Finish up reading 4.5 and 4.6. We will state the the rank plus nullity theorem in terms of linear transformations, but most of the applications will be interms of null spaces and column spaces of matrices. As a change of pace, and a prelude to chapter 5, we will discuss section 4.9. You will discover that we have seen much of this material before; only the terminology "steady state vector" will be new. Do problems 4.5 PP #1, 2, and exercises #1, 2, 6*, 9-13, 14*, 19-21, 22*, 23-25, 26*, 29-32; do problems 4.6 PP #1-4, 1, 2*, 3, 5, 6, 7, 8*, 9, 17, 18, 24*, 25, 26, 30, 31. The starred problems will be due on Monday, 6 April. Do problems 4.9 PP #1, 2, Exercises #1, 3, 4*, 5-7, 9-11, 13, 14*, 16, 19. The starred problems will be due on Wed., 8 April.
  • April 6-April 12. Markov chains (concl.), eigenvalues and eigenvectors Read sections 5.1 and 5.2. Think about where you have seen the eigenvalue 1 before! In 5.1 do PP #1, 2, Ex #1-12, 15, 16*, 17, 18*, 21-25, 26*, 29, 33; in 5.2 do the PP, then Ex #1-4, 9-11, 18*, 19, 21, 22. Starred problems will be due on Friday, 17 April, but be working on them before the exam! It would be a good idea to read 5.3 and 5.4--you will see that except for the similarity concept, and the applications to difference equations, as seen for example in Ex. #24, 25, 27 of 5.2, we have already done a lot of the work.Example 3 beginning on p. 321 is very nice, as are examples 4 and 5.
  • April 13-April 19. Diagonalization, similarity, Test #3 Read sections 5.3 and 5.4. In 5.3, do PP #1 (use diagonalization, not A^8 on your calculator), 2, 3, Ex #1, 2, 5, 6*, 7, 8*, 9-12, 15, 16* (use the info given right after #6). In 5.4, do PP #1, 2, Ex #8*, 13, 14, 17-19, 23, 25, 26*. The starred problems will be due on Monday, April 27.
  • April 20-April 27. Inner products, orthogonality, orthonormal sets, orthogonal and symmetric matrices Read 6.1, 6.2, 7.1. In 6.1, do PP #1-4, Ex #9, 10, 15-20, 22, 25, 26*, 27-29, 30*, 31. In 6.2, do PP #1, 3, Ex #1, 2, 7-9, 10*, 17, 18, 20, 21, 23a-d, 24a-c,e, 25 (with m = n), 26*, 29, 30, 32. To see the bigger picture, read/skim 6.7, and do PP #1, 2, Ex #13, 14, 15, 21, 23, 25, and 26 (the last will be a bonus problem). This material will not be on the final exam. In 7.1, do PP #1, 2, Ex #7, 8, 10, 13-17, 22*, 23, 24, 25, 26, 27, 28*, 29, 31, 32*. The starred problems will be due on Monday, April 27. I will provide a solution sheet for these problems at that time.

    • Exams
  • FIRST EXAM: Monday, February 9 (class #12) . This will cover sections 1.1-1.8/1.9 (content through Monday, Feb. 2). I will attempt to administer Quizzes 2 and 3 before the exam, and another quiz or two before the "drop deadline" of Monday, 23 February.
  • SECOND EXAM: Friday, March 6 (class #23). This will cover 1.9, the part of 1.10 that we did, 2.1-2.3, 3.1-3.3, and 4.1 (on sets that are or are not vector spaces).
  • THIRD EXAM: Wednesday, April 15 (class #37). The content will be drawn from 4.1-4.6, 4.9, 5.1-5.2 (restricted to topics that were covered in class through Friday, 10 April).
  • FINAL EXAM: Thursday, April 30. There will be two parts, as announced on the syllabus. Part B (40 points) will cover material not covered by the first three exams, while Part A (120 points) will recap the part of the course that was previously covered by the first two exams.
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    If you have already practiced the quizzes and exams from this semester, see my old ones from 2006 and 2002, including some solutions, as well as those of some of my colleagues for the current and past semesters (it looks like Professor Lu did the Honors 550 last spring, for example).

    • Last modified: April 20, 2009