Linear Algebra

Spring 2023

Instructor: Matt Kerr
Office: Cupples I, Room 114
e-mail: matkerr [at] math.wustl.edu
Office Hours: 8-9 M (on Zoom), 3-4 W (in office)

Prerequisites:

Math 310 or permission of instructor. Math 309 is not an explicit prerequisite, but familiarity with basic topics from matrix theory (matrix operations, linear systems, row reduction, Gaussian elimination) is expected, as these will be covered quickly.

Class Schedule:

Lectures are on Monday, Wednesday, and Friday from 11-11:50 in Crow 206, beginning Wednesday Jan. 18 and ending with a final exam review class on Friday Apr. 28. Spring Break is March 12-18.

Midterm Exam 1: Wednesday March 1 (in class), covering up to IV.B
Midterm Exam 2: Monday April 3 (in class), covering up to VI.C
Final Exam: Tuesday, May 9, 10:30am-12:30pm, in Crow 206

Regarding missed exams, see the Grading Policy section below. Calculators aren't allowed, but the exams will not be computationally heavy.

Textbook:

I encourage you to obtain a copy of:

Friedberg, Insel, and Spence, "Linear Algebra (4th Ed.)", Pearson, 2003;

but the lecture notes posted below will be our primary text.

Course Syllabus:

Proof-based introduction to the linear algebra of finite-dimensional vector spaces. A brief outline:

I. Linear Systems
II. Vector Spaces
III. Linear Transformations
IV. Determinants
V. Eigenvectors and Eigenvalues
VI. Canonical Forms
VII. Inner Products and Spectral Theory

Assignments:

There will be a weekly homework due by 8 PM on Wednesday, taken from the exercises at the end of each section of the Lecture Notes below. You will submit these through Gradescope. I am available for help in office hours. (Regarding late homework, cf. the Grading Policy below.)

Problem Set 1: (I.A) #3,4; (I.B) #3,4; (I.C) #5,6 [due Tuesday Jan. 24]
Problem Set 2: (I.D) #3,4,5,6,7; (II.A) #1,2,3,5; (II.B) #1,2,3,5 [due Tuesday Jan. 31]
Problem Set 3: (II.C) #3; (II.D) #1,2; (II.E) #1,4,5 [due Tuesday Feb. 7]
Problem Set 4: (III.A) #1-8; (III.B) #1-4 [due Wednesday Feb. 15]
Problem Set 5: (III.D) #1-6; (IV.A) #1,3,4; (IV.B) #1-4 [due Wednesday Feb. 22]
No HW due Wednesday Mar. 1 (prepare for exam)
Problem Set 6: (IV.C) #1-5; (V.A) #1,2,4,5; (V.B) #1(i,iii),2,3,5 [due Wednesday Mar. 8]
Problem Set 7: (V.B) #6,7; (V.C) #2,3,4,5; (VI.A) #1,2,3,4,5 [due Thursday Mar. 23]
Problem Set 8: (VI.B) #1-4; (VI.C) #1-7 [due Thursday Mar. 30]
Problem Set 9: (VI.D) #1,2,4; (VI.E) #1,2,3,5,6 [due Thursday Apr. 13]
Problem Set 10: (VII.A) #1,2,3,4; (VII.B) #1,2,3; (VII.C) #1,3,4 [due Thursday Apr. 20]
Problem Set 11: (VII.D) #1,2,3,4,5,6,7; (VII.E) #2,4,5 [due Friday Apr. 28]

Graders: Tiana Johnson (j.tiana [at] wustl.edu), Cesar Meza (c.j.meza [at] wustl.edu)

Lecture Notes:

Will be posted below no later than the day of the lecture. The hope is that this will make note-taking optional.

I. Linear Systems

A. Geometry of linear systems
B. Gauss-Jordan elimination
C. Matrix algebra
D. Solving linear systems via row-reduction

II. Vector Spaces

A. Vector spaces and subspaces
B. Basis and dimension
C. A "nice" basis for subspaces of Rⁿ
D. There is exactly one rref matrix row-equivalent to any A
E. Coordinates

III. Linear Transformations

A. Linear transformations without matrices
B. Linear transformations in terms of matrices
C. Linear functionals I: Derivations and Jacobians
D. Linear functionals II: the dual space

IV. Determinants

A. Alternating multilinear functionals
B. Computational methods for determinants
C. Properties and applications of determinants

V. Eigenvectors and Eigenvalues

A. Linear dynamical systems
B. Mechanics and theory of diagonalization
C. Complex eigenvalues and other topics

VI. Canonical Forms

A. On the minimal polynomial of a transformation
B. Normal form of a (polynomial) matrix
C. Rational canonical form
D. Generalized eigenspaces
E. Jordan normal form

VII. Inner Products and Spectral Theory

A. Review of orthogonality
B. Bilinear forms
C. Inner products
D. The spectral theorem
E. The singular value decomposition
F. Fourier series

Grading Policy:

Homework is worth 40% of your final grade; Midterm Exam 1 and Midterm Exam 2 are worth 15% each; and the Final Exam is worth 30%. I will drop your lowest 2 homework scores. Grades will be kept track of on Canvas.

Curving and grade scale: In the event that the average score on any exam is less than 75%, all exam scores will be adjusted upward by adding a constant to everyone's score (so that the average is 75%). No adjustment is made if the average is above 75%. The grade scale is as follows:

A+	A	A-	B+	B	B-	C+	C	C-	D	F
TBA	90+	[85,90)	[80,85)	[75,80)	[70,75)	[65,70)	[60,65)	[55,60)	[50,55)	[0,50)

The Pass/Fail policy is that you must get at least a C- to earn a "Pass".

If you have to miss an hour exam for a legitimate reason, you will be given a makeup exam. Of course verified illness and serious family emergency are legitimate reasons. (For the final exam, those are the only acceptable reasons.) Regarding other conflicts, e-mail me as soon as you know about them.

In general, credit will be given for late homework only in the event of illness or emergency.

You may discuss homework with other students (calculators are of course also allowed), but you should not have duplicate solutions. This link takes you to the standard university policies on academic integrity.