Math 350: Linear Algebra


Instructor: Joseph Palmer
Email: j.palmer at rutgers.edu
Schedule: 12:00-1:20pm Mondays and Thursdays in SEC-212
Textbook: Linear Algebra 4th ed. by Friedberg, Insel, and Spence (ISBN: 0-13-008451-4; ISBN13: 9780130084514)
Office Hours: Wednesdays 10am-noon in my office

Sakai: You can check your grades in this course with sakai. Please let me know as early as possible if you think there is an error on the sakai gradebook.

About this course:

This course is a proof-based approach to linear algebra. The topics we discuss will be very similar to those covered in Math 250, but these topics will be presented in a completely different manner. We will (hopefully) cover nearly the entire textbook, which encompasses the following five general topics:

Grade Breakdown:

20% Homework (and possibly quizzes)
20% Midterm 1
20% Midterm 2
40% Final Exam

Solutions and notes:

Fields (the definition of a field)
Determinant notes (to supplement Ch 4 from your text)

Homework:

Homework Assignments:

Homework 1 - due Thursday, Jan 25th in class:
Section 1.2: 1, 2, 4(b), 7, 13, 17, 21
Notes: Remember to be very formal in these proofs. Mention which properties (V1)-(V8) you are using and be sure to be careful about "for all" and "there exists". (there are no proofs or arguments needed in problem 1)
Suggested problems (not required to be turned in)
Section 1.1: 4
Section 1.2: 8, 9, 10, 11, 12, 14, 15, 20

Homework 2 - due Thursday, Feb 1st in class:
Section 1.3: 1, 2(d), 4, 5, 6, 8ab, 11, 18
(Again you don't need any proofs for problem 1)
Suggested problems (not required)
Section 1.3: 3, 7, 12, 13, 14, 17, 30, 31

Homework 3 - due Thursday, Feb 8th in class:
Section 1.4: 1c (and PROVE your answer for 1c), 5g, 8, 10, 12
Section 1.5: 7 (and prove answer), 9
Suggested problems (not required)
Section 1.4: 2, 3, 4, 13, 15, 16
Section 1.5: 1, 3, 5, 14, 20

Homework 4 - due Thursday, Feb 15th in class:
Section 1.6: 4, 13, 21, 26 (include proof, hint: find a basis)
Section 2.1: 10, 12, 15, 16
Suggested problems (not required)
Section 1.6: 1, 11, 15, 19, 22, 23, 24, 28, 29, 35
Section 2.1: 2, 6, 8, 9, 11, 14, 17, 18, 19, 21

Homework 5 - due Thursday, Feb 22nd in class:
Section 2.2: 3, 5d
Section 2.3: 9, 11
Suggested problems (not required):
Section 2.2: 5, 8, 9
Section 2.3: 1, 3, 12
Section 2.4: 2, 3, 4, 6, 9, 13, 14, 15, 16

Midterm 1 will be on Feb 26th, it will cover 1.1-1.6 and 2.1-2.5 (and maybe 2.6)
(notice that while there haven't been problems from 2.4 and 2.5 on the homework yet, there have been suggested problems about these sections)

Homework 6 - due Thursday, March 8th in class:
Section 2.4: 6, 15
Section 2.5: 10
Section 2.6: 2 (for each part if the answer is "no" then briefly state why)
Section 3.1: 6
Suggested problems (not required)
Section 2.5: 2, 5, 9
Section 2.6: 1abc
Section 3.1: 1

Homework 7 - due Thursday, March 22nd in class:
Section 3.2: 2ac (briefly explain), 7
Section 3.3: 2ae, 3ae, 4a, 5, 8a, 10
Section 3.4: 2ab
Suggested problems (not required)
Section 3.2: 2, 3, 4, 17
Section 3.3: 2, 3, 4, 7, 8, 9
Section 3.4: 2, 5

Homework 8 - due Thursday, March 29th in class:
There are notes posted about the determinant at the top of this page, use the definition of determinant from those notes to do the following problems from your textbook
Section 4.2: 2, 14, 26
Section 4.3: 10, 12, 15*, 16
Suggested problems (not required)
Section 4.2: 30
Section 4.3: 11, 14, 20
*for problem 15 recall A and B are similar if A = C^(-1)BC for some invertible C

Midterm 2 will be on Monday, April 9th, it will cover 3.1-3.4, chapter 4, and 5.1-5.2.
(as before, notice there have been no homework assignments on 5.1 and 5.2 yet, but homework 9 (posted below) includes these sections)

Homework 9 - due Thursday, April 12th in class:
Section 5.1: 3a, 4e, 7a, 12, 20
Section 5.2: 3ab, 8, 12
Suggested problems (not required)
Section 5.1: 1, 2, 3, 4, 8, 9, 21
Section 5.2: 1a-g, 2, 3, 11

Homework 10 - due Thursday, April 19th in class:
Section 6.1: 2, 9, 10, 16b, 17
Section 6.2: 2bc, 4, 6
Suggested problems (not required)
Section 6.1: 1, 3, 5, 6, 8
Section 6.2: 1, 2, 3, 5, 8, 11

Homework 11 - due Thursday, April 26th in class:
Section 6.2: 15, 16, 21
Section 6.3: 2ac, 3a, 7
Suggested problems (not required)
Section 6.3: 2, 3, 8, 11

Extra suggested problems:
Section 6.4: 2, 3, 11
Section 6.6: 5, 7bcdefg

FINAL EXAM IS ON THURSDAY MAY 3rd
The final will cover everything we have talked about in class so far, with extra emphasis placed on Chapter 6. It is good to be familiar with the sorts of problems on Midterm 1 and Midterm 2 and the assigned and suggested homework problems for all the sections.

Tentative Schedule:

This will be updated throughout the semester to stay as accurate as possible.

           Date       Sections       Topics
1:Th 1/18 1.1, 1.2 Introduction and Vector spaces
2:M 1/22 1.2 First properties of vector spaces
3:Th 1/25 1.3 Subspaces (HW1 due)
4:M 1/29 1.4 Linear combinations, span
4:Th 2/1 1.5 Linear independence (HW2 due)
5:M 2/5 1.6 Bases and dimension
5:Th 2/8 1.6 Bases and dimension (HW3 due)
6:M 2/12 2.1 Linear transformations
7:Th 2/15 2.2 Matrices (HW4 due)
8:M 2/19 2.3, 2.4 Composition of linear maps, Isomorphisms
9: Th 2/22 2.5, (2.6) Change of Coordinates (and dual spaces) (HW5 due)
 M 2/26 Midterm 1: Chapters 1 and 2
10:Th 3/1 Discuss exams, do examples, and catch up
11:M 3/5 3.1 Elementary matrices, rank, and inverses
12:Th 3/8 3.2, 3.3 Systems of linear equations I (HW6 due)
13:M 3/19 3.4 Systems of linear equations II (existence and uniqueness)
14:Th 3/22 Chap 4 Introducing the determinant (HW7 due)
15:M 3/26 Chap 4 Properties of det
16:Th 3/29 5.1 Eigenvalues and eigenvectors (HW8 due)
17:M 4/2 5.2 Diagonalizability
 Th 4/5 5.4 Invariant subspaces and Cayley-Hamilton Theorem (HW9 due)
 M 4/9 Midterm 2: Chapters 3, 4, 5
18:Th 4/12 6.1 Inner product spaces (or maybe 7.1?)
19:M 4/16 6.2 Gram-Schmidt
20:Th 4/19 6.3 Adjoints (HW10 due)
21:M 4/23 6.4 Normal/Self-Adjoint operators
22:Th 4/26 6.6 The Spectral Theorem (HW11 due)
M 4/30 Review and catch up
 Th 5/3 FINAL EXAM (SEC 212, 8-11am)