Math 416, Abstract Linear Algebra
Spring 2018
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, 4th edition, 600 pages, Pearson 2002.
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 are due at the beginning of class,
typically on a Friday. 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: February 14, March 14,
and April 18.
Final exam: The combined final exam for sections C13 and
D13 of Math 416 will be held on Monday, May 7, from 710pm in
1000 Lincoln Hall.
Missed exams: There will 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.
 Jan 17

Introduction. Section 1.1 of [FIS].
 Jan 19

Vectors spaces. Section 1.2 of [FIS].
 Jan 22

Subspaces. Section 1.3 of [FIS].
 Jan 24

Linear combinations and systems of
equations. Section 1.4 of [FIS] and Section
SSLE of
[B].
 Jan 26

Using matrices to encode and solve
linear systems.
Section RREF
of [B].
HW 1 due.
Solutions.
 Jan 29

Row echelon form and Gaussian
elimination.
Section RREF
of [B].
 Jan 31

Solution spaces to linear systems.
Section TSS
of [B].
 Feb 2

Linear dependence and independence.
Section 1.5 of [FIS].
HW 2 due.
Solutions.
 Feb 5

Basis and dimension.
Section 1.6 of [FIS].
 Feb 7

Basis and dimension, part 2.
Section 1.6 of [FIS].
 Feb 9

Basis, dimension, and linear systems.
HW 3 due.
Solutions.
 Feb 12

Intro to linear transformations.
Section 2.1 of [FIS].
 Feb 14

Midterm the First.
Handout, exam,
and solutions.
(Practice exam with
solutions.)
 Feb 16

The Dimension Theorem.
Section 2.1 of [FIS].
 Feb 19

Encoding linear transformations as
matrices. Section 2.2 of [FIS].
 Feb 21

Composing linear transformations
and matrix multiplication. Section 2.3 of [FIS].
 Feb 23

More on matrix multiplication.
Section 2.3 of [FIS].
HW 4 due.
Solutions.
 Feb 26

Isomorphisms and invertibility.
Section 2.4 of [FIS].
 Feb 28
 Matrices:
invertibility and rank. Section 2.4 of [FIS] and Sections
MINM and
CRS of [B].
 Mar 2

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

Introduction to determinants. Section 4.1 of [FIS].
 Mar 7

Definition of the determinant. Section 4.2 of [FIS].
 Mar 9

The determinant and row
operations. Section 4.2 of [FIS].
HW 6 due.
Solutions.
 Mar 12

Elementary matrices and the
determinant. Sections 3.1 and 4.3 of [FIS].
 Mar 14

Midterm the Second.
Handout, exam,
and solutions.
(Practice exam with
solutions.)
 Mar 16

Determinants and volumes. Section
4.3 of [FIS].
 Mar 17
 Spring Break starts.
 Mar 25
 Spring Break ends.
 Mar 26

Diagonalization and eigenvectors.
Section 5.1 of [FIS].
 Mar 28

Finding eigenvectors.
Sections 5.1 and 5.2 of [FIS].
 Mar 30

Diagonalization Criteria.
Section 5.2 of [FIS].
HW 7 due.
Solutions.
 Apr 2

Proof of the Diagonalization
Criteria. Section 5.2 of [FIS].
 Apr 4

Introduction to Markov Chains.
Section 5.3 of [FIS].
 Apr 6

Convergence of Markov Chains.
Section 5.3 of [FIS].
HW 8 due.
Solutions.
 Apr 9

Inner products.
Section 6.1 of [FIS].
 Apr 11

Inner products and orthogonality.
Sections 6.1 and 6.2 of [FIS].
 Apr 13

GramSchmidt and friends.
Section 6.2 of [FIS].
HW 9 due.
Solutions.
 Apr 16

Orthogonal complements and
projections. Sections 6.2 and 6.3 of [FIS].
 Apr 18

Midterm the Third.
Handout,
exam,
and solutions.
Practice exam
(solutions).
 Apr 20

Projections and adjoints.
Section 6.3 of [FIS].
 Apr 23

Normal and selfadjoint operators.
Section 6.4 of [FIS].
 Apr 25

Diagonalizing selfadjoint
operators.
Section 6.4 of [FIS].
HW 10 due.
Solutions.
 Apr 27

Orthogonal and unitary operators;
connections to quantum mechanics.
Section 6.5 of [FIS].
 Apr 30

Dealing with nondiagonalizable
matrices.
Section 6.7 and 7.1 of [FIS].
 May 2

Linear approximation, diagonalizing
symmetric matrices, and the second derivative test.
HW 11 due.
Solutions.
 May 7

Combined final exam from 710pm in 1000 Lincoln Hall.
Handout,
exam,
and solutions.
Practice exam
(solutions).