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 (Winter 2025):

Time Location Zoom Meeting ID Zoom Password
TTh 12:30-13:00 ESB 4127 and on Zoom 684 2674 1683 101671

F 10:30-11:30 My office and on Zoom

Time	Location	Zoom Meeting ID	Zoom Password
TTh 12:30-13:00	ESB 4127 and on Zoom	684 2674 1683	101671
F 10:30-11:30	My office and on Zoom

Classes: TTh 14:00-15:30 at LASR 211; see schedule below.
Syllabus
(Rough) lecture notes
(Almost) all information will be posted here; see Canvas for homework submission and solutions and the link to the Piazza discussion board.

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.

Problem Set 1 (LyX, TeX), due 14/1/2025. Solutions
Problem Set 2 (typo corrected), (LyX, TeX), due 23/1/2025. Solutions.
Problem Set 3, (LyX, TeX), due 3/2/2025. Solutions.
Problem Set 4, now due 10/2/2025. Solutions.
Problem Set 5, (LyX, TeX), due 17/2/2025. Solutions.
Problem Set 6, (LyX, TeX), due 4/3/2025.
Problem Set 7, (LyX, TeX), due 11/3/2025.
Problem Set 8, due 18/3/2025.

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 (postponed)
5	T 4/2	Examples Extension of Scalars		iPad notes
5	Th 6/2	\Sym^n and \wedge^n	§29,§30	PS4 (postponed); Feedback form
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
	T 4/3	Cayley--Hamilton	N 2.3	PS6 due; iPad notes
	Th 6/3	Jordan Blocks Nilpotent Jordan Form	§57, N 2.4	iPad notes
8	T 11/3	Vector Norms	§86, N 3.1	PS7 due
8	Th 13/3	Matrix Norms Power Method	§87, N 3.2 N 3.3
9	T 18/3	Completeness	N 3.4	PS8 due
9	Th 20/3	Series	N 3.4
10	T 25/3	Power series The Resolvent	N 3.5 N 3.6	PS9 due
10	Th 27/3	Holomorphic calculus	N 3.7
11	T 1/4	Composition	N 3.7	PS10 due
11	Th 3/4	Review
12	T 8/4	Review		PS11 due
	TBA	Final exam

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Clarification: the writings on these pages are generally my own creations (to which I own the copyright), and are made available for traditional academic reuse. If you wish to republish substantial portions (including in "derivative works") please ask me for permission. The material is expressly excluded from the terms of UBC Policy 81.

Last modified Wednesday March 12, 2025