Course Content

The textbook for the course is Linear Algebra Done Right by Sheldon Axler. We will follow it reasonably closely, although not verbatim, and we will not cover all material. Axler has also prepared videos summarizing the book. They are available free online at https://linear.axler.net/LADRvideos.html.

Approximate content of lectures:

• January 6: Introduction. Abstraction. Field C of complex numbers. LADR 1.A.
• January 10: Rⁿ and Cⁿ. Definition of vector space. LADR 1.A and 1.B.
• January 13: Vector subspaces. Sums of vector subspaces. LADR 1.C. whiteboards
• January 17: Spanning sets. LADR 2.A. whiteboards
• January 18: Linear independence. LADR 2.A. whiteboards
• January 20: Bases and dimension. LADR 2.B and 2.C.
• January 24: Bases and dimension. LADR 2.B and 2.C.
• January 25: Linear transformations, definition and examples. LADR 3.A.
• January 27: Image and kernel of a linear transformation. LADR 3.B.
• January 31: Surjective, injective, and invertible linear transformations. LADR 3.B and 3.D.
• February 1: Isomorphisms. LADR 3.D.
• February 3: Products of vector spaces. LADR 3.E.
• February 7: Products and quotients of vector spaces. LADR 3.E.
• February 8: Quotients of vector spaces. LADR 3.E.
• February 10: Quotients of vector spaces. LADR 3.E.
• February 15: The dual of a vector space. LADR 3.F.
• February 17: The dual of a vector space. LADR 3.F.
• February 28: Summarizing the course to date. Abstract row-echelon reduction.
• March 1: Midterm exam.
• March 3: Invariant subspaces. LADR 5.A.
• March 7: Eigenvalues and eigenvectors. Eigenbases and diagonalizable matrices. LADR 5.A and 5.C.
• March 8: Eigenvalues and generalizations. LADR 5.B.
• March 10: Existence of eigenvectors. LADR 5.B.
• March 14: Block matrices and invariant subspaces. LADR 5.A.
• March 15: Block upper triangular matrices. Restriction and quotient operators. LADR 5.A.
• March 17: Upper triangular matrices. LADR 5.B.
• March 21: Eventual kernel and eventual image. LADR 8.A.
• March 22: Generalized eigenspaces. LADR 8.A and 8.B.
• March 24: Decomposition of operators. Characteristic polynomials. LADR 8.B and 8.C.
• March 28: Characteristic polynomials. LADR 8.C.
• March 29: Complexification of real vector spaces. LADR 9.A.
• March 31: Characteristic polynomials of real operators. LADR 9.A.
• April 4: Decomposition of real operators. LADR 9.A.
• April 5: The trace of a matrix. LADR 10.A.

Homework assignments

Assignments should be submitted to theojf@dal.ca as a single PDF attachment, and can be neatly handwritten or typeset. Please make sure that your last name appears in the file name (and not just, say, "Homework 1").

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. The homeworks are designed to stretch you, and it is not expected that you will answer every question. The "100%" equivalent is listed below.

1. Assignment 1 (TeX source), due January 21. Solutions (TeX source). Score for 100%: 80/100.
2. Assignment 2 (TeX source), due February 1. Solutions (TeX source). Score for 100%: 50/60.
3. Assignment 3 (TeX source), due February 9. Solutions (TeX source). Score for 100%: 50/60.
4. Assignment 4 (TeX source), due February 17. Solutions (TeX source). Score for 100%: 40/60.
5. Assignment 5 (TeX source), due March 18. Solutions (TeX source). Score for 100%: 60/60.
6. Assignment 6 (TeX source), due March 25. Solutions (TeX source). Score for 100%: 60/60.
7. Assignment 7 (TeX source), due April 5. Solutions (TeX source). Score for 100%: 50/60.

Exams

The midterm exam will be held in-class on Tuesday, March 1. It will cover the first three chapters of Linear Algebra Done Right. Whereas homework assignments are primarily there to stretch your understanding and guide your learning, midterms primarily exist to test mastery of material. The exam will consist of six very short questions and one slightly longer one.

• Practice midterm and solutions.
• Midterm exam and solutions.

As set by the registrar, the final exam will be held in-person on Saturday, April 9, 7pm in Dunn 101. The exam will be similar in length, style, and difficulty to the midterm exam.

• Practice final and solutions.
• Final exam and solutions. Note: Although originally intended to be scored out of 40 points, the 100% mark on the final exam was set to 35, due to confusion over the notation "T^-1(w)" in question 4.