Math 416, Abstract Linear Algebra

Fall 2022

Course Description

This is a rigorous proof-oriented course in linear algebra. Topics include vector spaces, linear transformations, determinants, eigenvectors and eigenvalues, inner product spaces, Hermitian matrices, and Jordan Normal Form.

Prerequisites: Math 241 required with Math 347 strongly recommended.

Required text: Friedberg, Insel, and Spence, Linear Algebra, 5th edition, 608 pages, Pearson 2019.

The 4th edition, published in 2002, is also fine, as any differences in problem numbers will be noted in the assignments. Both the Math Library and Grainger Engineering Library have copies of the 4th and 5th editions on in-library reserve.

Supplementary text: Especially for the first quarter of the course, I will also refer to the free text:

Breezer, A First Course in Linear Algebra, Version 3.5 (2015). Available online or as a downloadable PDF file.

Course Policies

Overall grading: Your course grade will be based on homework (16%), three in-class midterm exams (18% each), and a comprehensive final exam (30%). You can view all of your scores in the online gradebook.

Weekly homework: These will typically be due on Friday. They are to be turned in on paper at the start of the class period. If you are unable to attend due to illness or quarantine, you can email me a PDF single file with a scan of your HW; if using your phone/tablet, please use an app designed for this purpose, such as Abobe Scan (iOS, Android). Late homework will not be accepted; however, your lowest two homework grades will be dropped, so you are effectively allowed two infinitely late assignments. Collaboration on homework is permitted, nay encouraged. However, you must write up your solutions individually and understand them completely.

In-class midterms: These three 50 minute exams will be held in our usual classroom on the following Wednesdays: September 21, October 19, and November 16.

Final exam: There will be a combined final exam for sections B13 and C13 of Math 416. The date and location of the final exam will be announced by the Registrar on October 13; until then, do not make plans to depart prior to Saturday, December 17.

Missed exams: There will typically be no make-up exams. Rather, in the event of a valid illness, accident, or family crisis, you can be excused from an exam so that it does not count toward your overall average. I reserve final judgment as to whether an exam will be excused. All such requests should be made in advance if possible, but in any event no more than one week after the exam date.

Cheating: Cheating is taken very seriously as it takes unfair advantage of the other students in the class. Penalties for cheating on exams, in particular, are very high, typically resulting in a 0 on the exam or an F in the class.

Disabilities: Students with disabilities who require reasonable accommodations should see me as soon as possible. In particular, any accommodation on exams must be requested at least a week in advance and will require a letter from DRES.

James Scholar/Honors Learning Agreements/4th credit hour: These are not offered for these sections of Math 416. Those interested in such credit should enroll in a different section of this course.

Detailed Schedule

Includes scans of my lecture notes and the homework assignments. Here [FIS] and [B] refer to the texts by Friedberg et al. and Breezer respectively.

Aug 22
Introduction. Section 1.1 of [FIS].
Aug 24
Vectors spaces. Section 1.2 of [FIS].
Aug 26
Subspaces. Section 1.3 of [FIS].
Aug 29
Linear combinations and systems of equations. Section 1.4 of [FIS] and Section SSLE of [B].
Aug 31
Using matrices to encode and solve linear systems. Section RREF of [B].
Sep 2
Row echelon form and Gaussian elimination. Section RREF of [B]. HW 1 due. Solutions.
Sep 5
Labor Day. No class.
Sep 7
Solution spaces to linear systems. Section TSS of [B].
Sep 9
Linear dependence and independence. Section 1.5 of [FIS]. HW 2 due. Solutions.
Sep 12
Basis and dimension. Section 1.6 of [FIS].
Sep 14
Basis and dimension, part 2. Section 1.6 of [FIS].
Sep 16
Basis, dimension, and linear systems. HW 3 due. Solutions.
Sep 19
Intro to linear transformations. Section 2.1 of [FIS].
Sep 21
Midterm the First. Handout, exam, and solutions. (Practice exam with solutions.)
Sep 23
The Dimension Theorem. Section 2.1 of [FIS].
Sep 26
Encoding linear transformations as matrices (draft notes). Section 2.2 of [FIS].
Sep 28
Composing linear transformations and matrix multiplication (draft notes). Section 2.3 of [FIS].
Sep 30
More on matrix multiplication (draft notes). Section 2.3 of [FIS]. HW 4 due.
Oct 3
Isomorphisms and invertibility (draft notes). Section 2.4 of [FIS].
Oct 5
Matrices: invertibility and rank (draft notes). Section 2.4 of [FIS] and Sections MINM and CRS of [B].
Oct 7
Changing coordinates (draft notes). Section 2.5 of [FIS]. HW 5 due.
Oct 10
Introduction to determinants (draft notes). Section 4.1 of [FIS].
Oct 12
Definition of the determinant (draft notes). Section 4.2 of [FIS].
Oct 14
The determinant and row operations (draft notes). Section 4.2 of [FIS]. HW 6 due.
Oct 17
Elementary matrices and the determinant (draft notes). Sections 3.1 and 4.3 of [FIS].
Oct 19
Midterm the Second.
Oct 21
Determinants and volumes (draft notes). Section 4.3 of [FIS].
Oct 24
Diagonalization and eigenvectors (draft notes). Section 5.1 of [FIS].
Oct 26
Finding eigenvectors (draft notes). Sections 5.1 and 5.2 of [FIS].
Oct 28
Diagonalization Criteria (draft notes). Section 5.2 of [FIS]. HW 7 due.
Oct 31
Proof of the Diagonalization Criteria (draft notes). Section 5.2 of [FIS].
Nov 2
Introduction to Markov Chains (draft notes). Section 5.3 of [FIS].
Nov 4
Convergence of Markov Chains (draft notes). Section 5.3 of [FIS]. HW 8 due.
Nov 7
Inner products (draft notes). Section 6.1 of [FIS].
Nov 9
Inner products and orthogonality (draft notes). Sections 6.1 and 6.2 of [FIS].
Nov 11
Gram-Schmidt and friends (draft notes). Section 6.2 of [FIS]. HW 9 due.
Nov 14
Orthogonal complements and projections (draft_notes). Sections 6.2 and 6.3 of [FIS].
Nov 16
Midterm the Third.
Nov 18
Projections and adjoints (draft notes). Section 6.3 of [FIS].
Nov 19
Thanksgiving Break starts.
Nov 26
Thanksgiving Break ends.
Nov 28
Normal and self-adjoint operators (draft notes). Section 6.4 of [FIS].
Nov 30
Diagonalizing self-adjoint operators (draft notes). Section 6.4 of [FIS]. HW 10 due.
Dec 2
Orthogonal and unitary operators; connections to quantum mechanics (draft). Section 6.5 of [FIS].
Dec 5
Dealing with nondiagonalizable matrices (draft notes). Section 6.7 and 7.1 of [FIS].
Dec 7
Linear approximation, diagonalizing symmetric matrices, and the second derivative test (draft notes). HW 11 due.
Dec ??
Combined final exam. Date and time TBA, but before Dec 17.