Math 412: Advanced Linear Algebra

Winter Term 2025

General Information

Office: MATX 1112, 604-827-3031
Email: "lior" (at) Math.UBC.CA (please include the course number in the subject line, if applicable)
Office hours (Fall 2024): after class (W 10:00-10:30 at the ESB atrium, W 14:00-14:30 at WW IRC Atrium), M 10:30-11:30 in my office, and by appointment.

Classes: TTh 14:00-15:30 at LASR 211; see schedule below.
Syllabus
(Rough) lecture notes

This is a second course in linear algebra, intended for honours students. There is no required textbook. The books by Roman and Halmos are both very good, cover most of the material, and are available in PDF form from the publisher for anyone on the UBC network; a lot of the material can also be found in any "abstract algebra" textbook. More details may be found in the syllabus.

References

Roman, Advanced Linear Algebra, available from SpringerLink
Halmos, Finite-dimensional Vector Spaces, available from SpringerLink
Coleman, Calculus on Normed Vector Spaces, Chapter 1 (on SpringerLink)
Higham, Functions of Matrices, available from SIAM
[Your favorite author], Abstract Algebra

Problem Sets

Problem sets are due at 14:00 Vancouver time (start of class) on the date indicated. Submission is on Canvas, and I expect typeset solutions.
- The source files for the problem sets rely on my macro file. Feel free to use it for your submissions as well.
Solutions (only) are stored on a secure website; registered students can access them after first logging on to Canvas.
Problem set grade statistics.

For your edification

Lecture-by-Lecture information

Section numbers marked § are in Halmos [2], section numbers marked N are in the course notes above.

Warning: the following information is tentative and subject to change at any time

Week	Date	Material	Reading	Notes
1	T 7/1	Introduction	§1,§2
1	Th 9/1	Direct sum and product	§19,§20	Note on infinite dimensions
2	T 14/1	(continued)		PS1 due
2	Th 16/1	Quotients	§21,§22
3	T 21/1	Duality	§13,§15
3	Th 23/1	(continued) Bilinear forms	§23	PS2 due
4	T 28/1	Tensor products	§24,§25
4	Th 30/1	(continued)		PS3 due
5	T 4/2	Examples Extension of Scalars		Feedback form
5	Th 6/2	\Sym^n and \wedge^n	§29,§30	PS4 due
6	T 11/2	(continued)
6	Th 13/2	Motivation The minimal polynomial	N2.1 N2.2	PS5 due
	Feb 17-21	Midterm break
	T 25/2	Midterm Exam
7	Th 27/2	Generalized eigenspaces	N2.3
7	T 4/3	Cayley--Hamilton	N 2.3	PS6 due
8	Th 6/3	Jordan Blocks Nilpotent Jordan Form	§57, N 2.4
8	T 11/3	Nilpotent Jordan form Jordan canonical form	§57, N 2.4 §58, N 2.5	PS7 due
9	Th 13/3	Vector Norms	§86, N 3.1
9	T 18/3	Matrix Norms Power Method	§87, N 3.2 N 3.3	PS8 due
10	Th 20/3	Completeness	N 3.4
10	T 25/3	Series	N 3.4	PS9 due
11	Th 27/3	Power series The Resolvent	N 3.5 N 3.6
11	T 1/4	Holomorphic calculus	N 3.7	PS10 due
12	Th 3/4	Composition	N 3.7
12	T 8/4	Review		PS11 due
	TBA	Final exam

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Last modified Thursday December 05, 2024