# 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 (Fall 2019): Mondays 12:30-14:00 IBLC Learning Lounge, Thursdays 11:00-12:00 in my office

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

## Midterm Exam

• The exam will take place in-class on Thursday, October 24. The material for it is the "constructions" chapter of the course (lectures up to and including Thursday, October 10.
• Here is a previous midterm.

## 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.
3. Problem Set 3, due 26/9/2019.

## 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 PS2 due
4 T 24/9 (continued)
Bilinear forms

§23

Th 26/9 Tensor products §24,§25 PS3 due
5 T 1/10 (continued)
Th 3/10 Extension of Scalars   PS4 due; Feedback form
6 T 8/10 \Sym^n and \wedge^n §29,§30
Th 10/10 (continued)   PS5 due
7 T 15/10 Motivation
The minimal polynomial
N2.1
N2.2

Th 17/10 Generalized eigenspaces N2.3 PS6 due
8 T 22/10 Cayley--Hamilton
Jordan Blocks
N 2.3
§57, N 2.4

Th 24/10 Midterm exam
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 PS7 due
T 5/11 Matrix Norms
Power Method
§87, N 3.2
N 3.3

10 Th 7/11 Completeness N 3.4 PS8 due
T 12/11 Series N 3.4
11 Th 14/11 Power series
The Resolvent
N 3.5
N 3.6
PS9 due
T 19/11 Holomorphic calculus N 3.7
12 Th 21/11 Composition N 3.7 PS10 due
T 26/11
13 Th 28/11 Review   PS11 due
TBA Final exam

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