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. Solutions.
  7. Problem Set 7, due 11/3/2025. Solutions.
  8. Problem Set 8, due 18/3/2025. Solutions.
  9. Problem Set 9, due 25/3/2025. Solutions.
  10. Problem Set 10, due 1/4/2025. Solutions.
  11. Problem Set 11, due 8/4/2025. Solutions.

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
Jordan form examples
8 T 11/3 Vector Norms
Matrix norms		 §86, N 3.1
§87, N 3.2		 PS7 due;
iPad notes
Th 13/3 Power Method N 3.3 iPad notes
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 Vignette: Linear groups    
12 T 8/4 Representation Theory   PS11 due
  T 15/4 Final exam 15:30--18:30 @ BUCH B215
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Last modified Tuesday April 08, 2025