Math 416, Abstract Linear Algebra

Spring 2016

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, 4th edition, 600 pages, Pearson 2002.

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%).

Weekly homework: These are due at the beginning of class, typically on a Friday. 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 17, March 16, and April 20.

Final exam: There will be a combined final exam for sections B13 and C13 of Math 416, which will be held on Friday, May 6 from 1:30-4:30 in Psychology 23.

Missed exams: There will 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 20
Introduction. Section 1.1 of [FIS].
Jan 22
Vectors spaces. Section 1.2 of [FIS].
Jan 25
Subspaces. Section 1.3 of [FIS].
Jan 27
Linear combinations and systems of equations. Section 1.4 of [FIS] and Section SSLE of [B].
Jan 29
Using matrices to encode and solve linear systems. Section RREF of [B]. HW 1 due. Solutions.
Feb 1
Row echelon form and Gaussian elimination. Section RREF of [B].
Feb 3
Solution spaces to linear systems. Section TSS of [B].
Feb 5
Linear dependence and independence. Section 1.5 of [FIS]. HW 2 due. Solutions.
Feb 8
Basis and dimension, part 1. Section 1.6 of [FIS].
Feb 10
Basis and dimension, part 2. Section 1.6 of [FIS].
Feb 12
Basis, dimension, and linear systems. HW 3 due. Solutions.
Feb 15
Intro to linear transformations. Section 2.1 of [FIS].
Feb 17
Midterm the First. Handout. Solutions.
Feb 19
The Dimension Theorem. Section 2.1 of [FIS].
Feb 22
Encoding linear transformations as matrices. Section 2.2 of [FIS].
Feb 24
Composing linear transformations and matrix multiplication. Section 2.3 of [FIS].
Feb 26
More on matrix multiplication. HW 4 due. Section 2.3 of [FIS]. Solutions.
Feb 29
Isomorphisms and invertibility. Section 2.4 of [FIS].
Mar 2
Matrices: invertibility and rank. Section 2.4 of [FIS] and Sections MINM and CRS of [B].
Mar 4
Changing coordinates. Section 2.5 of [FIS]. HW 5 due. Solutions.
Mar 7
Introduction to determinants. Section 4.1 of [FIS].
Mar 9
Definition of the determinant. Section 4.2 of [FIS].
Mar 11
The determinant and row operations. Section 4.2 of [FIS]. HW 6 due. Solutions.
Mar 14
Elementary matrices and the determinant. Sections 3.1 and 4.3 of [FIS].
Mar 16
Midterm the Second. Handout. Solutions.
Mar 18
Determinants and volumes. Section 4.3 of [FIS].
Mar 19
Spring Break starts.
Mar 27
Spring Break ends.
Mar 28
Diagonalization and eigenvectors. Section 5.1 of [FIS].
Mar 30
Finding eigenvectors. Sections 5.1 and 5.2 of [FIS].
Apr 1
Diagonalization Criteria. Section 5.2 of [FIS]. HW 7 due. Solutions.
Apr 4
Proof of the Diagonalization Criteria. Section 5.2 of [FIS].
Apr 6
Matrix powers and Markov Chains. Section 5.3 of [FIS].
Apr 8
Convergence of Markov Chains. Section 5.3 of [FIS]. HW 8 due. Solutions.
Apr 11
Inner products. Section 6.1 of [FIS].
Apr 13
Inner products and orthogonality. Sections 6.1 and 6.2 of [FIS].
Apr 15
Gram-Schmidt and friends. Section 6.2 of [FIS]. HW 9 due. Solutions.
Apr 18
Orthogonal complements and projections. Sections 6.2 and 6.3 of [FIS].
Apr 20
Midterm the Third. Handout. Solutions.
Apr 22
Projections and adjoints. Section 6.3 of [FIS].
Apr 25
Normal and self-adjoint operators. Section 6.4 of [FIS].
Apr 27
Diagonalizing self-adjoint operators. Section 6.4 of [FIS]. HW 10 due. Solutions.
Apr 29
Orthgonal and unitary operators. Section 6.5 of [FIS].
May 2
Dealing with nondiagonalizable matrices. Section 6.7 and 7.1 of [FIS].
May 4
Linear approximation, diagonalizing symmetric matrices, and the second derivative test. HW 11 due. Solutions.
May 6
Final exam from 1:30 - 4:30 pm in Psychology 23. Handout. Solutions.

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