Math 416, Abstract Linear Algebra
Fall 2022
Course Description
This is a rigorous prooforiented course in linear algebra. Topics
include vector spaces, linear transformations, determinants,
eigenvectors and eigenvalues, inner product spaces, Hermitian
matrices, and Jordan Normal Form.
Prerequisites: Math 241 required with Math 347 strongly
recommended.
Required text: Friedberg, Insel, and Spence, Linear
Algebra, 5th edition, 608 pages, Pearson 2019.
The 4th edition, published in 2002, is also fine, as any
differences in problem numbers will be noted in the assignments. Both
the Math Library and Grainger Engineering Library have copies of
the
4th and 5th editions on inlibrary reserve.
Supplementary text: Especially for the first quarter of the
course, I will also refer to the free text:
Breezer, A First Course in Linear Algebra, Version 3.5 (2015).
Available online or as a
downloadable PDF file.
Course Policies
Overall grading: Your course grade will be based on
homework (16%), three inclass midterm exams (18% each), and a
comprehensive final exam (30%). You can view all of your scores in the
online gradebook.
Weekly homework: These will typically be due on Friday.
They are to be turned in on paper at the start of the
class period. If you are unable to attend due to illness or
quarantine, you can email me a PDF single file with a scan of your HW;
if using your phone/tablet, please use an app designed for this
purpose, such as Abobe Scan (iOS,
Android).
Late homework will not be accepted; however, your lowest two homework
grades will be dropped, so you are effectively allowed two infinitely
late assignments. Collaboration on homework is permitted, nay
encouraged. However, you must write up your solutions individually and
understand them completely.
Inclass midterms: These three 50 minute exams will be held in
our usual classroom on the following Wednesdays: September 21, October 19,
and November 16.
Final exam: The combined final exam for sections B13 and
C13 of Math 416 will be held on Monday, December 12, from 811am in
1092 Lincoln Hall. If you have a conflict, please email me to arrange
for a conflict exam.
Missed exams: There will typically be no makeup
exams. Rather, in the event of a valid illness, accident, or
family crisis, you can be excused from an exam so that it does not
count toward your overall average. I reserve final judgment as to
whether an exam will be excused. All such requests should be
made in advance if possible, but in any event no more than one
week after the exam date.
Cheating: Cheating is taken very seriously as it takes
unfair advantage of the other students in the class. Penalties for
cheating on exams, in particular, are very high, typically resulting
in a 0 on the exam or an F in the class.
Disabilities: Students with disabilities who require
reasonable accommodations should see me as soon as possible. In
particular, any accommodation on exams must be requested at least a
week in advance and will require a letter from DRES.
James Scholar/Honors Learning Agreements/4th credit hour: These
are not offered for these sections of Math 416. Those interested in such
credit should enroll in a different section of this course.
Detailed Schedule
Includes scans of my lecture notes and the homework assignments.
Here [FIS] and [B] refer to the texts by Friedberg et al. and Breezer
respectively.
 Aug 22

Introduction. Section 1.1 of [FIS].
 Aug 24

Vectors spaces. Section 1.2 of [FIS].
 Aug 26

Subspaces. Section 1.3 of [FIS].
 Aug 29

Linear combinations and systems of
equations. Section 1.4 of [FIS] and Section
SSLE of
[B].
 Aug 31

Using matrices to encode and solve
linear systems.
Section RREF
of [B].
 Sep 2

Row echelon form and Gaussian
elimination.
Section RREF
of [B].
HW 1 due.
Solutions.
 Sep 5

Labor Day. No class.
 Sep 7

Solution spaces to linear
systems.
Section TSS
of [B].
 Sep 9

Linear dependence and
independence.
Section 1.5 of [FIS].
HW 2 due.
Solutions.
 Sep 12

Basis and dimension.
Section 1.6 of [FIS].
 Sep 14

Basis and dimension, part 2.
Section 1.6 of [FIS].
 Sep 16

Basis, dimension, and
linear systems.
HW 3 due.
Solutions.
 Sep 19

Intro to linear
transformations.
Section 2.1 of [FIS].
 Sep 21

Midterm the First.
Handout, exam,
and solutions.
(Practice exam with
solutions.)
 Sep 23

The Dimension Theorem.
Section 2.1 of [FIS].
 Sep 26

Encoding linear transformations as
matrices. Section 2.2 of [FIS].
 Sep 28

Composing linear
transformations and matrix multiplication. Section 2.3 of [FIS].
 Sep 30

More on matrix multiplication.
Section 2.3 of [FIS].
HW 4 due.
Solutions.
 Oct 3

Isomorphisms and invertibility.
Section 2.4 of [FIS].
 Oct 5
 Matrices:
invertibility and rank. Section 2.4 of [FIS] and Sections
MINM and
CRS of [B].
 Oct 7

Changing coordinates.
Section 2.5 of [FIS].
HW 5 due.
Solutions.
 Oct 10

Introduction to determinants. Section 4.1 of [FIS].
 Oct 12

Definition of the
determinant. Section 4.2 of [FIS].
 Oct 14

The determinant and row
operations. Section 4.2 of [FIS].
HW 6 due.
Solutions.
 Oct 17

Elementary matrices and the
determinant. Sections 3.1 and 4.3 of [FIS].
 Oct 19

Midterm the Second.
Handout,
exam,
and solutions.
(Practice exam with
solutions.)
 Oct 21

Determinants and volumes. Section
4.3 of [FIS].
 Oct 24

Diagonalization and
eigenvectors.
Section 5.1 of [FIS].
 Oct 26

Finding eigenvectors.
Sections 5.1 and 5.2 of [FIS].
 Oct 28

Diagonalization Criteria.
Section 5.2 of [FIS].
HW 7 due.
Solutions.
 Oct 31

Proof of the Diagonalization
Criteria. Section 5.2 of [FIS].
 Nov 2

Introduction to Markov Chains.
Section 5.3 of [FIS].
 Nov 4

Convergence of Markov Chains.
Section 5.3 of [FIS].
HW 8 due.
Solutions.
 Nov 7

Inner products.
Section 6.1 of [FIS].
 Nov 9

Inner products and
orthogonality.
Sections 6.1 and 6.2 of [FIS].
 Nov 11

GramSchmidt and friends.
Section 6.2 of [FIS]. Fourier
example. HW 9 due.
Solutions.
 Nov 14

Orthogonal complements and
projections. Sections 6.2 and 6.3 of [FIS].
 Nov 16

Midterm the Third.
Handout, exam,
and solutions.
Practice exam
(solutions).
 Nov 18

Projections and adjoints.
Section 6.3 of [FIS].
 Nov 19
 Thanksgiving Break starts.
 Nov 26
 Thanksgiving Break ends.
 Nov 28

Normal and selfadjoint
operators.
Section 6.4 of [FIS].
 Nov 30

Diagonalizing selfadjoint
operators.
Section 6.4 of [FIS].
HW 10 due.
Solutions.
 Dec 2

Orthogonal and unitary operators;
connections to quantum mechanics.
Section 6.5 of [FIS].
 Dec 5

Dealing with nondiagonalizable
matrices.
Section 6.7 and 7.1 of [FIS].
 Dec 7

Linear approximation, diagonalizing
symmetric matrices, and the second derivative test.
HW 11 due.
Solutions.
 Dec 12

Combined final exam. 8am in 1092 Lincoln Hall.
Handout.
Practice exam
(solutions).