Advanced Linear Algebra

Resources

These are resources I created while I was TAing MATH 2055U (Advanced Linear Algebra) at Ontario Tech University in the Winter 2024 term. They should still be applicable to equivalent courses at other universities. The resources are intended for students who have completed a first course on linear algebra and are in a proof-based course. The main textbook used throughout these materials is Linear Algebra Done Right by Sheldon Axler, 3e. To get a variety, I also will reference Linear Algebra Done Wrong by Sergei Treil. Topics will be presented in the order of Axler. Those not using Axler or Treil will still find these resources helpful. Students in this course may also find the materials produced for a first course on linear algebra to be useful; these resources can be found here.

Video Examples

  1. Vector spaces and subspaces
    Note: For more videos on this topic, see the videos listed as 13.X and 14.X in Linear Algebra
    1. Question 1.1
    2. Question 1.2
    3. Question 1.3
    4. Question 1.4
  2. Basis and dimension
    1. Question 2.1
    2. Question 2.2
    3. Question 2.3
    4. Question 2.4
    5. Question 2.5
    6. Question 2.6
  3. Inner product spaces
    1. Question 3.1
    2. Question 3.2
    3. Question 3.3
    4. Question 3.4
    5. Question 3.5
  4. Linearity
    1. Question 4.1
    2. Question 4.2

    5. Problem Sets

    1. Proof and Vector Space Basics
    2. Basis and Dimension
    3. Inner Product Spaces
    4. Linear Transformations
    5. Fun with Isomorphisms
    6. Midterm Review
    7. Change of Basis
    8. Matrix Products and the Exponential
    9. Eigenstuff
    10. Similarity and Diagonalization
    11. Singular Value Decomposition

    Lecturette Recordings

    Note: The quality of the recordings may not be ideal. They are not meant to replace in-class attendance. They should primarily serve as a supplemental aid to students. Note: recording 6 is omitted as it was a midterm review.
    1. Basis and Dimension
    2. Inner Product Spaces
    3. Linear Transformations
    4. Fun with Isomorphisms
    1. Change of Basis
    2. Matrix Product and the Exponential
    3. Eigenstuff
    4. Similarity and Diagonalization
    5. Singular Value Decomposition

    Lecturette Notes

    Note: The notes provided here are handwritten and scanned. The scans aren't perfect, but I hope are enough for students of mine to refer back to. Note: notes 6 are omitted as it was a midterm review.
    1. Proof and Vector Space Basics
    2. Basis and Dimension
    3. Inner Product Spaces
    4. Linear Transformations
    5. Fun with Isomorphisms
    1. Change of Basis
    2. Matrix Products and the Exponential

    You may find some of the challenge questions in my resources for the first course in linear algebra helpful. These challenge questions can be accessed here.

    For those who want to go a step further or prefer learning by diagrams, here is a fantastic resource you can use by Prof. Pawel Sobocinski.