Honors Linear Algebra
Mathematics 350 Spring 2016

Prof. Lepowsky (640:350:H1)

The course is strongly based on Math 250. However, we'll work axiomatically, starting from the abstract notions of vector space and of linear transformation. Much of the homework and many of the exam and quiz problems will require you to write precise proofs, building on your proof-writing experience in Math 300. From this more abstract viewpoint, we'll be developing linear algebra far beyond Math 250, with new insight and new applications.

Class attendance is very important. A lot of what we do in class will involve collective participation.

We will cover the topics indicated in the syllabus below, but the dates that we cover some of the topics might be adjusted during the semester, depending on class discussion, etc. Such adjustments, along with the almost-weekly homework assignments, will be announced in class and also posted on this webpage, so be sure to check this webpage regularly.

HW is due on most Wednesdays. There is no HW due on the two midterm-exam Wednesdays.

Note that we will cover significant material from all the chapters in the book, Chapters 1-7, but we will cover Chapter 7 before Chapter 6. This is because the material in Chapter 7 is a natural continuation of the material in Chapter 5 on the theory of eigenvalues, eigenvectors and diagonalizability. Chapter 6 also concerns eigenvalues, eigenvectors and diagonalizability, but this time, based on a generalization of the theory of dot products.

Quizzes will be given at the ends of a few class sessions. The dates of these quizzes, and the topics covered, will be announced in advance.

Final exam: Tuesday, May 10, 2016, 12:00 noon - 3:00 PM. (The date and time of the final exam are linked to the class meeting time.) UPDATE: The final exam room will be the regular lecture room.

Grading policy: First midterm exam: 100 points; Second midterm exam: 100 points; Homework and quizzes: 100 points; Final exam: 200 points (Total: 500 points).

Tentative Course Syllabus

This information will be updated throughout the semester.

Week Lecture dates  Sections   Topics
1 1/20 Chapter 1 Abstract vector spaces
21/25, 1/27Chapter 1 Subspaces, span of subsets, linear independence
32/1, 2/3 Chapter 1 Bases and dimension
42/8, 2/10 Chapter 2 Linear transformations, matrix representation
52/15, 2/17 Chapter 2 Composition, invertibility
62/22, 2/24 Ch. 1-2  Review and Exam 1
72/29, 3/2 Ch. 2 & 3  Change of basis, dual spaces; rank and systems of linear equations. Note: We've already essentially covered Chapter 3, starting from the beginning of the course, since we've been reviewing and using the methods of Math 250.
83/7, 3/9 Chapter 4  Determinants and their properties
93/21, 3/23 Chapter 5  Eigenvalues, eigenvectors, diagonalizability
103/28, 3/30 Ch. 5 & 7  Invariant subspaces and the Cayley-Hamilton Theorem, Jordan canonical form
114/4, 4/6 Chapter 7  Jordan canonical form
124/11, 4/13 Ch. 3-5, 7  Review and Exam 2
134/18, 4/20 Chapter 6  Inner product spaces
144/25, 4/27 Chapter 6  Normal and self-adjoint operators, unitary and orthogonal operators
155/2   Review
Tuesday, May 10 Noon-3 PM Final Exam (in the regular lecture room)

Homework Assignments and Additional Comments

This information will be updated throughout the semester.

HW due     HW problems (due Wednesdays)
Jan. 27 1.2 #9, 13; 1.3 #8(a),(e), 18, 19. For the Sec. 1.2 problems, quote which axioms among (VS 1)-(VS 8) on page 7 you are invoking in all your steps. Also, in this HW and thoughout the entire course, be sure to explicitly and correctly use the quantifiers "for every" and "there exist(s)," as we did in class.
Feb. 3 1.4 #4(a), 5(g) (Hint: Rewrite the 2 x 2 matrices as column vectors), 12; 1.5 #2(e), 9; 1.6 #3(a),(b), 11. Always use the method of augmented matrices and rref, as we've reviewed in detail in class from Math 250, whenever this method is relevant. In your writeup of problem 1.4 #4(a), explicitly write the arrow notation (like for instance -3r_2 + r_3 --> r_3) for each elementary row operation in your row reduction. For all the remaining problems, you don't need to write this arrow notation explicitly, and you can combine two or more elementary row operations in each step in your row reductions, if you do this carefully. Be sure that your row operations are actually *elementary* row operations. (For instance, r_2 + 3r_3 --> r_3 is *not* an elementary row operation.)

Recall that in this course, we are using the list of conditions in the statement of Thm. 1.3 as our official definition of "subspace," rather than the book's (equivalent) definition on p. 16, but recall that a subspace does in fact satisify the conditions in the book's definition on p. 16.

In general, throughout the course, write everything clearly and precisely.
Feb. 10 1.6 #15, 18, 20, 26; 2.1 #2, 3, 9(a)-(e), 13, 15, 20; 2.2 #2(a), 4, 5(d). Note: Since we didn't do Sec. 2.1 in class on Feb. 3, you can postpone handing in your solutions to the Sec. 2.1 and 2.2 problems until Monday, Feb. 15. Note: Use the idea of L_A (p. 92) in 2.1 #2, 3 and 2.2 #2(a)
Feb. 17 2.1 #17; 2.2 #2(c),(e),(g), 5(c),(g), 8, 10; 2.3 #3(a),(b), 4(d) [See 2.2 #5(d) for the notation T], 12
Mar. 2 Due after the exam: 2.3 #13; 2.4 #2(f), 9, 10; 2.5 #2(b), 3(d), 5, 6(c), 10. Note: Since we didn't do Sec. 2.5 in class on Feb. 22, this "Mar. 2" HW is due Mar. 7.
Mar. 9 2.6 #2, 3(a),(b), 4; 3.1 #4, 5, 6, 7; 3.2 #6(a); 3.3 #2(b), 3(b); 3.4 #5
Mar. 23 4.1 #3(a), 5, 6, 9, 11; 4.2 #7, 18, 20 (use elementary row or column operations for #18, 20); 4.3 #15, 24. Be sure to read and think about Appendix D, on the field of complex numbers, which you'll need in this assignment and which we'll be using later. You can skip the proof of the Fundamental Theorem of Algebra, Theorem D.4, but you should know its statement. Also, catch up if necessary!
Mar. 30 5.1 #3(d), 4(b),(f), 8, 12, 14; 5.2 #2(b),(d),(f), 3(d), 12
Apr. 6 4.3 #21; 5.4 #2, 3, 6(a),(b), 9 (for 6(a) and (b) only), 10 (for 6(a) and (b) only), 15
Apr. 25 Note that this last assignment is due Apr. 25, not Apr. 20. 7.1 #2(c), 3(a) (For each of these Sec. 7.1 problems, you can construct a cycle of generalized eigenvectors starting either at the beginning of the cycle or at the end of the cycle); 7.2 #2, 3; 6.1 #4, 8, 9, 11; 6.2 #2(d), 11, 18

Homework policy

You are allowed, and encouraged, to work together with other people in the class, if you wish, and also to ask me for hints, either by email or in class or in office hours. But your final HW writeups need to be your own, expressed in your own way and showing your own understanding. Also, nowadays it's easy to find solutions to problems in textbooks, including this book. But in order to really benefit from the HW (and in order to be able to do your best on the exams), you of course shouldn't seek out or use existing solutions while you're working on the HW problems.

Main 350 course page

Jim Lepowsky / lepowsky@math.rutgers.edu / Spring 2016