Math 416, Abstract Linear Algebra

Spring 2024

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, Pearson 2019 (hardcover) or 2022 (paperback).

The 4th edition, published in 2002, is also fine, as any differences in problem numbers will be noted in the assignments. Warning: The Bookstore initially had the wrong textbook for this class. Please check that you have purchased the correct book. Also, Amazon has the much cheaper paperback edition. Grainger Engineering Library has two copies of the 5th edition on in-library reserve; there are also copies of the 4th edition available for checkout at the Main Library.

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: February 14, March 20, and April 17.

Final exam: There will be a combined final exam for sections C13 and D13 of Math 416, which will be held Thursday May 9 from 8am to 11am in 120 Architecture Building.

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.

Jan 17
Introduction. Section 1.1 of [FIS].
Jan 19
Vectors spaces. Section 1.2 of [FIS].
Jan 22
Subspaces. Section 1.3 of [FIS].
Jan 24
Linear combinations and systems of equations. Section 1.4 of [FIS] and Section SSLE of [B].
Jan 26
Using matrices to encode and solve linear systems. Section RREF of [B]. HW 1 due.
Jan 29
Row echelon form and Gaussian elimination. Section RREF of [B].
Jan 31
Solution spaces to linear systems. Section TSS of [B].
Feb 2
Linear dependence and independence. Section 1.5 of [FIS]. HW 2 due. Solutions.
Feb 5
Basis and dimension. Section 1.6 of [FIS].
Feb 7
Basis and dimension, part 2. Section 1.6 of [FIS].
Feb 9
Basis, dimension, and linear systems. HW 3 due. Solutions.
Feb 12
Intro to linear transformations. Section 2.1 of [FIS].
Feb 14
Midterm the First. Handout, exam, and solutions. (Practice exam with solutions.)
Feb 16
The Dimension Theorem. Section 2.1 of [FIS].
Feb 19
Encoding linear transformations as matrices. Section 2.2 of [FIS].
Feb 21
Composing linear transformations and matrix multiplication. Section 2.3 of [FIS].
Feb 23
More on matrix multiplication. Section 2.3 of [FIS]. HW 4 due. Solutions.
Feb 26
Isomorphisms and invertibility. Section 2.4 of [FIS].
Feb 28
Matrices: invertibility and rank. Section 2.4 of [FIS] and Sections MINM and CRS of [B].
Mar 1
Changing coordinates. Section 2.5 of [FIS]. HW 5 due. Solutions.
Mar 4
Introduction to determinants. Section 4.1 of [FIS].
Mar 6
Definition of the determinant. Section 4.2 of [FIS].
Mar 8
The determinant and row operations. Section 4.2 of [FIS]. HW 6 due. Solutions.
Mar 9
Spring Break starts.
Mar 17
Spring Break ends.
Mar 18
Elementary matrices and the determinant. Sections 3.1 and 4.3 of [FIS].
Mar 20
Midterm the Second. Handout, exam, and solutions. (Practice exam with solutions.)
Mar 22
Determinants and volumes. Section 4.3 of [FIS].
Mar 25
Diagonalization and eigenvectors. Section 5.1 of [FIS].
Mar 27
Finding eigenvectors. Sections 5.1 and 5.2 of [FIS].
Mar 29
Diagonalization Criteria. Section 5.2 of [FIS]. HW 7 due. Solutions.
Apr 1
Proof of the Diagonalization Criteria. Section 5.2 of [FIS].
Apr 3
Introduction to Markov Chains. Section 5.3 of [FIS].
Apr 5
Convergence of Markov Chains. Section 5.3 of [FIS]. HW 8 due. Solutions.
Apr 8
Inner products. Section 6.1 of [FIS].
Apr 10
Inner products and orthogonality. Sections 6.1 and 6.2 of [FIS].
Apr 12
Gram-Schmidt and friends. Section 6.2 of [FIS]. Fourier example. HW 9 due. Solutions.
Apr 15
Orthogonal complements and projections. Sections 6.2 and 6.3 of [FIS].
Apr 17
Midterm the Third. Handout, exam, and solutions. Practice exam (solutions).
Apr 19
Projections and adjoints. Section 6.3 of [FIS].
Apr 22
Normal and self-adjoint operators. Section 6.4 of [FIS].
Apr 24
Diagonalizing self-adjoint operators. Section 6.4 of [FIS]. HW 10 due. Solutions.
Apr 26
Orthogonal and unitary operators; connections to quantum mechanics. Section 6.5 of [FIS].
Apr 29
Dealing with nondiagonalizable matrices. Section 6.7 and 7.1 of [FIS].
May 1
Linear approximation, diagonalizing symmetric matrices, and the second derivative test. HW 11 due. Solutions.
May 9
Combined final exam. 8am-11am in room 120 of the Arhitecture Building. Handout. Solutions; Practice exam (solutions).