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: The combined final exam for sections B13 and C13 of Math 416 will be held on Monday, December 12, from 8-11am in 1092 Lincoln Hall. If you have a conflict, please email me to arrange for a conflict exam.

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. Section 2.2 of [FIS].
Sep 28
Composing linear transformations and matrix multiplication. Section 2.3 of [FIS].
Sep 30
More on matrix multiplication. Section 2.3 of [FIS]. HW 4 due. Solutions.
Oct 3
Isomorphisms and invertibility. Section 2.4 of [FIS].
Oct 5
Matrices: invertibility and rank. Section 2.4 of [FIS] and Sections MINM and CRS of [B].
Oct 7
Changing coordinates. Section 2.5 of [FIS]. HW 5 due. Solutions.
Oct 10
Introduction to determinants. Section 4.1 of [FIS].
Oct 12
Definition of the determinant. Section 4.2 of [FIS].
Oct 14
The determinant and row operations. Section 4.2 of [FIS]. HW 6 due. Solutions.
Oct 17
Elementary matrices and the determinant. Sections 3.1 and 4.3 of [FIS].
Oct 19
Midterm the Second. Handout, exam, and solutions. (Practice exam with solutions.)
Oct 21
Determinants and volumes. Section 4.3 of [FIS].
Oct 24
Diagonalization and eigenvectors. Section 5.1 of [FIS].
Oct 26
Finding eigenvectors. Sections 5.1 and 5.2 of [FIS].
Oct 28
Diagonalization Criteria. Section 5.2 of [FIS]. HW 7 due. Solutions.
Oct 31
Proof of the Diagonalization Criteria. Section 5.2 of [FIS].
Nov 2
Introduction to Markov Chains. Section 5.3 of [FIS].
Nov 4
Convergence of Markov Chains. Section 5.3 of [FIS]. HW 8 due. Solutions.
Nov 7
Inner products. Section 6.1 of [FIS].
Nov 9
Inner products and orthogonality. Sections 6.1 and 6.2 of [FIS].
Nov 11
Gram-Schmidt and friends. Section 6.2 of [FIS]. Fourier example. HW 9 due. Solutions.
Nov 14
Orthogonal complements and projections. Sections 6.2 and 6.3 of [FIS].
Nov 16
Midterm the Third. Handout, exam, and solutions. Practice exam (solutions).
Nov 18
Projections and adjoints. Section 6.3 of [FIS].
Nov 19
Thanksgiving Break starts.
Nov 26
Thanksgiving Break ends.
Nov 28
Normal and self-adjoint operators. Section 6.4 of [FIS].
Nov 30
Diagonalizing self-adjoint operators. Section 6.4 of [FIS]. HW 10 due. Solutions.
Dec 2
Orthogonal and unitary operators; connections to quantum mechanics. Section 6.5 of [FIS].
Dec 5
Dealing with nondiagonalizable matrices. Section 6.7 and 7.1 of [FIS].
Dec 7
Linear approximation, diagonalizing symmetric matrices, and the second derivative test. HW 11 due. Solutions.
Dec 12
Combined final exam. 8am in 1092 Lincoln Hall. Handout. Practice exam (solutions).