Course Content

The textbook for the class is Linear Algebra Done Right (LADR), fourth edition, by Sheldon Axler. This book is available from the Dalhousie Bookstore and from the author's webpage. The author has also recorded videos summarizing sections of the book.

Additional resources

• Essence of linear algebra, video series by 3blue1brown. (EoLA)
• Introduction to mathematical arguments, by Michael Hutchings. Helps with writing proofs. Go over the first three sections (skipping 3.2, if you prefer) and do the exercises at the end of Section 2. The appendix on set theory can also be an useful refresher.
• Linear Algebra Done Wrong, by Sergei Treil. Open access. (LADW)
• Linear Algebra, by Peter Selinger. Open access.
• Jupyter Guide to Linear Algebra, by Ben Vanderlei. Open access. Useful to see the concepts we see in class implemented in Python.

Approximate content of lectures

1. Jan 07: Introduction. Complex numbers. Rⁿ and Cⁿ. LADR 1A. EoLA 1
2. Jan 09: Abstract vector spaces. LADR 1B.
3. Jan 14: Subspaces. Spans. LADR 1C and 2A.
4. Jan 16: Linear dependence and independence. Bases. LADR 2A (cont.) and 2B. EoLA 2
5. Jan 21: Bases and coordinates. Dimension. LADR 2B (cont.) and 2C. LADW 2.8.
6. Jan 23: Linear transformations. LADR 3A. LADW 1.3.
7. Jan 28: Linear transformations and matrices. LADR 3C. EoLA 3 & 4
8. Jan 30: Null space and range. LADR 3B. End of the first midterm content.
9. Feb 04: Invertible linear transformations. LADR 3D.
10. Feb 06: How to use determinants. Midterm review. EoLA 6
11. Feb 11: First midterm test.
12. Feb 13: Isomorphisms. New vector spaces from old.
13. Feb 18: Reading week.
14. Feb 20: Reading week.
15. Feb 25: Zeroes of complex polynomials. Definition of eigenvalue & eigenvector. notes LADR 4 & 5A. EoLA 14
16. Feb 27: Complex operators have eigenvalues. notes LADR 5A (cont.) & 5B.
17. Mar 04: Minimal polynomial. Triangular matrices. notes LADR 5B (cont.) & 5C.
18. Mar 06: Eigenvalues & diagonal matrices. Diagonalization. notes LADR 5C (cont.) & 5D.
19. Mar 11: Change of basis & diagonalization. End of the second midterm content. resource EoLA 13.
20. Mar 13: Inner products & norms. LADR 6A. EoLA 9
21. Mar 18: Second midterm test.
22. Mar 20: Orthonormal bases. Gram-Schmidt procedure. LADR 6B
23. Mar 25: Self adjoint & normal operators. LADR 7A
24. Mar 27: Spectral theorem. LADR 7B
25. Apr 01. Isometries & unitary operators. LADR 7D
26. Apr 03: Positive operators or catch-up. LADR 7C

That is it! This is arguably all of "introductory" linear algebra. We skipped some sections of the book: 5E on commuting operators (important in quantum mechanics), 7E on singular value decomposition (important in machine learning), 8 on the Jordan normal form, and 9 on quadratic forms & tensor products.

Problem sets

While homework sets are designed to help you keep up to date to the course, problem sets aim to stretch your understanding. If ever a question doesn't say, assume that you should justify your answers. When explaining or justifying an answer, formulate your reasoning into well-written sentences and paragraphs. You are not expected to answer every question.

Problem sets are not graded, and as such they have no due date. You can submit them for feedback any time (in paper, via email, or during office hours), and you don't have to submit all the problems. Some of the problems might appear in the exams.

• Problem Set 1 (LaTeX source) - solutions (vector spaces and subspaces)
• Problem Set 2 (LaTeX source) - solutions (spans, linear independence, and bases)
• Problem Set 3 (LaTeX source) - solutions are in Brightspace (linear transformations)
• Problem Set 4 (LaTeX source) (isomorphisms & eigenvalue basics)
• Problem Set 5 (LaTeX source) (more eigenthings & diagonalization)

Exams

The midterm tests will be two-stage exams. Students will first complete and turn in an individual exam, and then will complete the same exam again working in small groups. The grade for these tests will be the largest of the following options:

• individual portion (80%) + group portion (20%)
• individual portion (100%)

Students writing the exam at the accessibility centre will only do the individual portion of the exam (which hence composes all of the grade).

• First midterm: February 11th, in class. solutions practice test (solutions)
• Second midterm: March 18th, in class.
• Final exam: April 16th, 3:30pm (location TBD).

Homework

Homework is assigned weekly through WebWork on the course Brightspace page. Sets will tipically open Sunday, end of day, and due on the Tuesday of the following week, end of day. (e.g. HW2 opens Jan 12 and is due Jan 21)

Applications of Linear Algebra

Many of concepts and theorems that we will see in this course are important to various applied science fields. Here we take the opportunity to highlight such applications. You can send your submission to daniel.teixeira@dal.ca, or submit it through Brightspace. Periodically, submissions will be posted for other students to learn about such applications. You can ask your instructor for topic suggestions.

Sample Submission