# Math 412: Advanced Linear Algebra

Fall Term 2019
Lior Silberman

## 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 2024): by appointment or
T 9:30-10:00ORCH 3009N/AN/A
Th 9:30-11:00ORCH 3009 and on Zoom694 6667 3745914585
F 11:45-13:00at PIMS and on Zoom676 1308 4912139267

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

2. Halmos, Finite-dimensional Vector Spaces, available from SpringerLink
3. Coleman, Calculus on Normed Vector Spaces, Chapter 1 (on SpringerLink)
4. Higham, Functions of Matrices, available from SIAM
5. [Your favorite author], Abstract Algebra

## Problem Sets

• Solutions (only) are stored on a secure website; registered students can access them after first logging on to Canvas.
1. Problem Set 1, due 12/9/2019. Solutions.
2. Problem Set 2, due 24/9/2019. Solutions.
3. Problem Set 3, due 1/10/2019. Solutions.
4. Problem Set 4, due 8/10/2019. Solutions.
5. Problem Set 5, due 15/10/2019. Solutions.
6. Problem Set 6, due 22/10/2019. Solutions.
7. Problem Set 7, due 5/11/2019. Solutions.
8. Problem Set 8, due 19/11/2019. Solutions.
9. Problem Set 9, due 19/11/2019. Solutions.
10. Problem Set 10, not for submission. Solutions.

## 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

1 Th 5/9 Introduction §1,§2
2 T 10/9 Direct sum and product §19,§20 Note on infinite dimensions
Th 12/9 (continued)   PS1 due
3 T 17/9 Quotients §21,§22
Th 19/9 Duality §13,§15
4 T 24/9 (continued)
Bilinear forms

§23
PS2 due
Th 26/9 Tensor products §24,§25
5 T 1/10 (continued)   PS3 due
Th 3/10 Extension of Scalars     Feedback form
6 T 8/10 \Sym^n and \wedge^n §29,§30 PS4 due
Th 10/10 (continued)
7 T 15/10 Motivation
The minimal polynomial
N2.1
N2.2
PS5 due
Th 17/10 Generalized eigenspaces N2.3
8 T 22/10 Cayley--Hamilton N 2.3 PS6 due
Th 24/10 Jordan Blocks
Nilpotent Jordan Form
§57, N 2.4
T 29/10 Nilpotent Jordan form
Jordan canonical form
§57, N 2.4
§58, N 2.5

9 Th 31/10 Vector Norms §86, N 3.1
T 5/11 Matrix Norms
Power Method
§87, N 3.2
N 3.3
PS7 due
10 Th 7/11 Completeness N 3.4
T 12/11 Series N 3.4
11 Th 14/11 Power series
The Resolvent
N 3.5
N 3.6

T 19/11 Holomorphic calculus N 3.7 PS8, PS9 due
12 Th 21/11 Composition N 3.7
T 26/11     PS10 due
13 Th 28/11 Review
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.