Math 412: Advanced Linear Algebra

Winter Term 2025
Lior Silberman

General Information

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

  1. Roman, Advanced Linear Algebra, available from SpringerLink
  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

  1. Problem Set 1 (LyX, TeX), due 14/1/2025. Solutions
  2. Problem Set 2 (typo corrected), (LyX, TeX), due 23/1/2025. Solutions.
  3. Problem Set 3, (LyX, TeX), due 3/2/2025. Solutions.
  4. Problem Set 4, now due 10/2/2025. Solutions.
  5. Problem Set 5, (LyX, TeX), due 17/2/2025. Solutions.
  6. Problem Set 6, (LyX, TeX), due 4/3/2025.
  7. Problem Set 7, (LyX, TeX), due 11/3/2025.
  8. 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  
Th 9/1 Direct sum and product §19,§20 Note on infinite dimensions
2 T 14/1 (continued)   PS1 due
Th 16/1 Quotients §21,§22  
3 T 21/1 Duality §13,§15  
Th 23/1 (continued)
Bilinear forms
 
§23
PS2 due
4 T 28/1 Tensor products §24,§25  
Th 30/1 (continued)   PS3 (postponed)
5 T 4/2 Examples
Extension of Scalars
  iPad notes
Th 6/2 \Sym^n and \wedge^n §29,§30 PS4 (postponed); Feedback form
6 T 11/2 (continued)    
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
Th 13/3 Matrix Norms
Power Method
§87, N 3.2
N 3.3
 
9 T 18/3 Completeness N 3.4 PS8 due
Th 20/3 Series N 3.4  
10 T 25/3 Power series
The Resolvent
N 3.5
N 3.6
PS9 due
Th 27/3 Holomorphic calculus N 3.7  
11 T 1/4 Composition N 3.7 PS10 due
Th 3/4 Review    
12 T 8/4 Review   PS11 due
  TBA Final exam    


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Last modified Wednesday March 12, 2025