Linear Algebra
Mathematics 350 Fall 2016

Prof. Lepowsky (640:350:01)

The course is heavily based on Math 250, and starting early in the course, we'll be reviewing and using the methods of 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 Thursdays. There is no HW due on the two midterm-exam Thursdays (one of which is actually Tuesday, Nov. 22, which has a Thursday class schedule).

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, Dec. 20, 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. Be sure to check this page regularly.

Week Lecture dates  Sections   Topics
1 9/8 Chapter 1 Abstract vector spaces
29/12, 9/15Chapter 1 Subspaces, span of subsets, linear independence
39/19, 9/22 Chapter 1 Bases and dimension
49/26, 9/29 Chapter 2 Linear transformations, matrix representation
510/3, 10/6 Chapter 2 Composition, invertibility
610/10, 10/13 Ch. 1-2  Review and Exam 1
710/17, 10/20 Ch. 2 & 3  Change of basis. Note: We have already essentially covered Chapter 3, starting from the beginning of the course, since we have been reviewing and using the methods of Math 250.
810/24, 10/27 Chapter 4  Determinants and their properties
910/31, 11/3 Chapter 5  Eigenvalues, eigenvectors, diagonalizability
1011/7, 11/10 Chapter 5  Invariant subspaces
1111/14, 11/17 Ch. 5 & 7  The Cayley-Hamilton Theorem, Jordan canonical form
1211/21, 11/22 (Note: 11/22 has Thursday class schedule.) Ch. 3-5, 7  Review and Exam 2
1311/28, 12/1 Ch. 7 & 6  Jordan canonical form, inner product spaces
1412/5, 12/8 Chapter 6  Normal and self-adjoint operators, unitary and orthogonal operators
1512/12   Review
Tuesday, December 20 Noon-3 PM. UPDATE: The final exam room will be the regular lecture room. Final Exam

Homework Assignments and Additional Comments

This information will be updated throughout the semester. Be sure to check this information regularly.

HW due     HW problems (due on most Thursdays)
HW #1, Sept. 15 Sec. 1.2 #2, 4(b),(f), 8, 9, 12, 17
(In #8 and #9: Explicitly quote each vector space axiom (among (VS 1)--(VS 8)) that you use in each step. In #8, explain why parentheses are not needed in the right-hand side of the formula. In #12, you need to show that all the parts of the definition of vector space hold.)
Sec. 1.3 #5, 8(a),(b), 11
Note: To prove that a certain subset of a vector space is a subspace, verify the three conditions in Theorem 1.3. To prove that a certain subset of a vector space is *not* a subspace, you only need to show that *one* of the three conditions in Theorem 1.3 fails.
HW #2, Sept. 22 Sec. 1.4 #4(a), 5(g) (Hint: You can rewrite the 2 x 2 matrices as column vectors if you want), 6, 9
Sec. 1.5 #2(d), 6
Sec. 1.6 #2(b), 3(b), 9
Important note: In most of the problems in HW #2, you'll need to set up an appropriate system of system of m linear equations in n variables, and then use the augmented-matrix procedure to find all the solutions, or to determine that the system is inconsistent. In HW #2, you need to indicate all of your elementary row operations, including the explicit arrow notation, that you use to reduce the augmented matrix to reduced row echelon form, resulting in a system of linear equations equivalent to the original system, as we did in an extended example in class. Then you need to interpret the unique solution, or the infinitely many solutions, or the inconsistency of the system, in order to answer the question. Sometimes you will need to determine whether a system Ax=b (written in matrix form) has a solution for *every* vector b in F^m. In this HW, you will be reviewing important material from Math 250; see Prof. Speer's review notes and the discussion below about elementary row operations, and be sure to consult the relevant material from the Math 250 text.
In 1.6 #2(b) and 3(b), you are allowed to quote Corollary 2(a) or 2(b) (pages 47-48), which allow you to conclude that an n-element spanning set in an n-dimensional vector space is automatically linearly independent (and therefore is a basis), and an n-element linearly independent set in an n-dimensional vector space automatically spans (and therefore is a basis). We won't prove this Corollary and the related theorems this week, but you can quote them when they're relevant.
Quiz 1, Sept. 26 Quiz 1, Monday, Sept. 26, will be on the Math 250 material reviewed in Prof. Speer's notes (see below). We've been using this material in class to answer questions about spanning, linear (in)dependence, and bases.

Each quiz will be about 20 minutes, at the end of a class period. Each quiz will count 10 points (the equivalent of five 2-point HW problems).
HW #3, Sept. 29 Sec. 2.1 #2, 3, 9(a),(b), 15
Sec. 2.2 #2(a),(b),(c), 4, 5(d)

IMPORTANT HINT: In several of these problems, observe that T is of the form L_A, which we discussed in detail in class (and note that L_A isn't defined until p. 92). When you need to find bases of N(L_A) and of R(L_A), use the relevant methods of Math 250 in order to find bases of Null(A) and Col(A).
HW #4, Oct. 6 Sec. 2.3 #3(a),(b), 4(d)
Sec. 2.4 #2(a),(b),(c),(d), 3(a),(c), 7(a)
HW #5, due date postponed to Oct. 24 Sec. 2.5 #2(b),(c), 3(c),(d), 5, 6(b),(c),(d)
HW #6, Oct. 27
Note: I'm adding Sec. 3.3 #2(b) and #3(b) to HW #6.
Sec. 3.2 #5(d),(e), 6(a) (Use the standard ordered basis, and Cor. 1 of Thm. 2.18, p. 102, and then use Thm. 2.14, p. 91, to compute the inverse of T)
Sec. 3.3 #2(b), 3(b)
Sec. 3.4 #6 (Use the Column Correspondence Property)
Note: Except for 3.2.6(a), this is further review of Math 250.
Quiz 2, Oct. 31 Quiz 2, Monday, Oct. 31, will be chosen from the same material you prepared for Exam 1, including the material in Exam 1 itself. Use this as a chance to review all the required proofs and HW problems.
HW #7, Nov. 3 Sec. 4.2 #7, 8, 14, 18 (Use elementary row operations)
Sec. 4.3 #12, 15, 24 (Hint: Call the given matrix A_n. The goal is to show that det(A_n + tI_n) is given by the formula on p. 578, in the answers for Sec. 4.3. Use mathematical induction on n. For this, first show that the formula holds for n=1. Then let n be greater than 1 and assume the induction hypothesis that the formula holds for n-1. Use this to prove the formula for n.)
HW #8, Nov. 10 Sec. 5.1 #3(c),(d), 4(b),(f), 8(a),(b), 9
Sec. 5.2 #2(b),(d),(f), 3(d), 12
HW #9, Nov. 17 Sec. 5.4 #2(a),(b),(c), 3, 6(a),(b), 9 (for 6(a),(b)), 10 (for 6(a),(b))
Note: I'm not assigning Sec. 5.4 #19, because you already did this problem, in Sec. 4.3 #24. This result is used in Sec. 5.4, p. 316.
HW #10, due date postponed to Dec. 5
Another note: Since on Dec. 1 we didn't have time to discuss Sec. 6.2 in class, you can hand in the Sec. 6.2 part of HW #10 on Dec. 8. But all of the rest of HW #10 is still due on Dec. 5. (What you'll need to know in Sec. 6.2 is basically review of Gram-Schmidt, etc., from Math 250.)
Sec. 7.1 #2(a),(c), 3(a),(b)
Sec. 7.2 4(a)
UPDATE: I've now omitted Sec. 7.2 #2 and #3. (Note that Sec. 7.2 #4(a) is similar to the Sec. 7.1 problems.)
Sec. 6.1 #2, 3, 9
Sec. 6.2 #2(a),(c)


Here is a link to
review notes on Math 250 prepared by Prof. Speer.
A comment concerning arrow notation for indicating elementary row operations: In these review notes, the notation (for example) r_3-->5r_3 is used to mean "multiply row 3 by 5," while in the Math 250 text, this same elementary row operation is described by the opposite arrow notation, namely, 5r_3-->r_3, which means "5 times row 3 replaces row 3" (and similarly for the arrow notations for all elementary row operations). Both directions of the arrow notations are standard. In this course, when you need to explicitly indicate arrow notations for elementary row operations, feel free to use either version of the arrow notation, as long as you are consistent in your choice of notation.
By the way, whenever you use elementary row operations, be sure that your row operations are actually *elementary* operations. For instance, r_2+3r_3-->r_3 is *not* an elementary row operation, since this indicates that row 2 plus 3 times row 3 replaces row 3 (in terms of the version of the arrow notation in the Math 250 text), while 3r_2+r_3-->r_3 *is* an elementary row operation,

Homework policy


In all of your homework writeups, you need to write clearly and precisely. In your proofs, you need to show your reasoning clearly.

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 of course in order to be able to do your best on the exams, you shouldn't seek out or use existing solutions while you're working on the HW problems.

A note on how HW will be graded: Each HW problem will be worth either 1 point or (usually) 2 points or sometimes more, depending on how routine or substantial the problem is. For instance, 1.2 #2 and 1.2 #4(b) and (f) will each be viewed as a 1-point problem, while 1.2 #8 will count as a 2-point problem. So your HW score each week will be your total out of a score that will vary somewhat from week to week. At the end of the course, your weekly totals will be added and your HW total will then be rescaled (depending on how many quizzes there will be).

Main 350 course page


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