Math 416, Abstract Linear Algebra
Spring 2016
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%).
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 17, March 16,
and April 20.
Final exam: There will be a combined final exam for
sections B13 and C13 of Math 416, which will be held on Friday, May 6
from 1:304:30 in Psychology 23.
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 20

Introduction. Section 1.1 of [FIS].
 Jan 22

Vectors spaces. Section 1.2 of [FIS].
 Jan 25

Subspaces. Section 1.3 of [FIS].
 Jan 27

Linear combinations and systems of
equations. Section 1.4 of [FIS] and Section
SSLE of
[B].
 Jan 29

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

Row echelon form and Gaussian
elimination.
Section RREF
of [B].
 Feb 3

Solution spaces to linear systems.
Section TSS
of [B].
 Feb 5

Linear dependence and independence.
Section 1.5 of [FIS].
HW 2 due.
Solutions.
 Feb 8

Basis and dimension, part 1.
Section 1.6 of [FIS].
 Feb 10

Basis and dimension, part 2.
Section 1.6 of [FIS].
 Feb 12

Basis, dimension, and linear systems.
HW 3 due. Solutions.
 Feb 15

Intro to linear transformations.
Section 2.1 of [FIS].
 Feb 17

Midterm the First. Handout. Solutions.
 Feb 19

The Dimension Theorem.
Section 2.1 of [FIS].
 Feb 22

Encoding linear transformations as
matrices. Section 2.2 of [FIS].
 Feb 24

Composing linear transformations
and matrix multiplication. Section 2.3 of [FIS].
 Feb 26

More on matrix multiplication.
HW 4 due.
Section 2.3 of [FIS]. Solutions.
 Feb 29

Isomorphisms and invertibility.
Section 2.4 of [FIS].
 Mar 2
 Matrices:
invertibility and rank. Section 2.4 of [FIS] and Sections MINM and CRS
of [B].
 Mar 4

Changing coordinates. Section 2.5 of [FIS].
HW 5 due.
Solutions.
 Mar 7

Introduction to determinants. Section 4.1 of [FIS].
 Mar 9

Definition of the determinant. Section 4.2 of [FIS].
 Mar 11

The determinant and row
operations. Section 4.2 of [FIS].
HW 6 due.
Solutions.
 Mar 14

Elementary matrices and the
determinant. Sections 3.1 and 4.3 of [FIS].
 Mar 16

Midterm the Second.
Handout. Solutions.
 Mar 18

Determinants and volumes. Section
4.3 of [FIS].
 Mar 19
 Spring Break starts.
 Mar 27
 Spring Break ends.
 Mar 28

Diagonalization and eigenvectors.
Section 5.1 of [FIS].
 Mar 30

Finding eigenvectors.
Sections 5.1 and 5.2 of [FIS].
 Apr 1

Diagonalization Criteria. Section 5.2 of [FIS].
HW 7 due.
Solutions.
 Apr 4

Proof of the Diagonalization
Criteria. Section 5.2 of [FIS].
 Apr 6

Matrix powers and Markov Chains.
Section 5.3 of [FIS].
 Apr 8

Convergence of Markov Chains. Section 5.3 of [FIS].
HW 8 due.
Solutions.
 Apr 11

Inner products.
Section 6.1 of [FIS].
 Apr 13

Inner products and orthogonality.
Sections 6.1 and 6.2 of [FIS].
 Apr 15

GramSchmidt and friends.
Section 6.2 of [FIS].
HW 9 due.
Solutions.
 Apr 18

Orthogonal complements and
projections. Sections 6.2 and 6.3 of [FIS].
 Apr 20

Midterm the Third.
Handout.
Solutions.
 Apr 22

Projections and adjoints.
Section 6.3 of [FIS].
 Apr 25

Normal and selfadjoint operators.
Section 6.4 of [FIS].
 Apr 27

Diagonalizing selfadjoint
operators.
Section 6.4 of [FIS].
HW 10 due.
Solutions.
 Apr 29

Orthgonal and unitary operators.
Section 6.5 of [FIS].
 May 2

Dealing with nondiagonalizable
matrices.
Section 6.7 and 7.1 of [FIS].
 May 4

Linear approximation, diagonalizing
symmetric matrices, and the second derivative test.
HW 11 due.
Solutions.
 May 6

Final exam from 1:30  4:30 pm in Psychology 23.
Handout.
Solutions.