What is linear algebra 2

One way to think about linear algebra is the study of linear transformations \(T:V \to V\) where \(V\) is a vector space. Another way to think about it is the study of \(n\) by \(n\) matrices, and all their associated data (rowspace, columnspace, nullspace, left-nullspace, eigenvectors, eigenvalues, etc.) In this course, we will try to take a mixed approach, sometimes studying linear algebra from an abstract perspective and sometimes from a concrete matrix perspective. Here are some of the topics you will see in this course:

  • Cayley-Hamilton theorem
  • Canonical forms
  • Inner product spaces and the spectral theorem
  • Singular value decomposition

What can you expect from this course

  • develop appreciation for the power of abstract thinking
  • work on your proof writing skills
  • work on your problem solving skills
  • apply linear algebra to solve problems

What I expect from you

  • skim notes before watching the videos
  • watch videos, and do the exercises at the end
  • if anything is not making sense, ask in class, or by e-mail

How to study

  • review definitions
  • do practice problems
  • invent your own problems
  • review proofs of theorems (do they make sense?)
  • summarize proofs of theorems

Assessment information

Tentatively (this may change based on feedback from AQAC)

  • 50% final exam
  • 20% first coursework exam
  • 20% second coursework exam
  • 10% assignments (5 assignments, drop the lowest mark)

Calendar

Week Topics Evaluation
1 Review of vector spaces, bases, dimension  
2 Direct sums Assignment 1 given
3 Block matrices Assignment 1 due
4 Polynomials Assignment 2 given
5 Linear transformations and matrices Assignment 2 due
6 Inner product spaces Exam 1
7 Diagonalization Assignment 3 given
8 Cayley-Hamilton Theorem Assignment 3 due
9 Canonical forms Assignment 4 given
10 Canonical forms II Exam 2
11 Singular value decomposition Assignment 4 due, Assignment 5 given
12 Quadric surfaces Assignment 5 due
13 revision