Math 416, Abstract Linear Algebra
Spring 2016
Course Description
This is a rigorous proof-oriented 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 in-class 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.
In-class 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 1:30-4:30 in Psychology 23.
Missed exams: There will be no make-up 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
-
Gram-Schmidt 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 self-adjoint operators.
Section 6.4 of [FIS].
- Apr 27
-
Diagonalizing self-adjoint
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.