Text: Linear Algebra, 4th ed., by Friedberg, Insel and Spence,
Prentice Hall, 2003, ISBN 0-13-008451-4.
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).
|1||9/8||Chapter 1||Abstract vector spaces|
|2||9/12, 9/15||Chapter 1||Subspaces, span of subsets, linear independence|
|3||9/19, 9/22||Chapter 1||Bases and dimension|
|4||9/26, 9/29||Chapter 2||Linear transformations, matrix representation|
|5||10/3, 10/6||Chapter 2||Composition, invertibility|
|6||10/10, 10/13||Ch. 1-2||Review and Exam 1|
|7||10/17, 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.|
|8||10/24, 10/27||Chapter 4||Determinants and their properties|
|9||10/31, 11/3||Chapter 5||Eigenvalues, eigenvectors, diagonalizability|
|10||11/7, 11/10||Chapter 5||Invariant subspaces|
|11||11/14, 11/17||Ch. 5 & 7||The Cayley-Hamilton Theorem, Jordan canonical form|
|12||11/21, 11/22 (Note: 11/22 has Thursday class schedule.)||Ch. 3-5, 7||Review and Exam 2|
|13||11/28, 12/1||Ch. 7 & 6||Jordan canonical form, inner product spaces|
|14||12/5, 12/8||Chapter 6||Normal and self-adjoint operators, unitary and orthogonal operators|
|Tuesday, December 20||Noon-3 PM. UPDATE: The final exam room will be the regular lecture room.||Final Exam|
|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
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)
Main 350 course page