# Math 524 — Linear Algebra — Syllabus

Some details of the syllabus may change between semesters; consult **your professor's** syllabus!

## Stable Course Components

The course components described in this section are mostly professor independent, and should not greatly change from semester-to-semester.

### Catalog Description

Vector spaces, linear transformations, orthogonality, eigenvalues and eigenvectors, normal forms for complex matrices, positive definite matrices, and congruence.

### Prerequisites

- Math 245 (Discrete Mathematics), and
- One of:
- Math 254 (Introduction to Linear Algebra), or
- Math 342A (Methods of Applied Mathematics), or
- Aerospace Engineering 280 (Methods of Analysis)

**Strongly Recommended:** at least one 300-level theoretical/formal Mathematics course, e.g. Mathematics 320,
or Mathematics 330; or 300-level class(es) in Engineering/Computer Science/Physics.

### Overview

The goal of this course is to provide a rigorous exposition of the fundamental components of linear algebra: Linear Transformation on Vector- and Inner Product Spaces; and show some of the foundational results — including Diagonalization (or Non-Diagonalization) using eigen-bases, the Gram-Schmidt Method, the Riesz Representation Theorem, the Real and Complex Spectral Theorems, the Cayley-Hamilton Theorem — and finally discuss the Singular Value Decomposition, as well as similarity transformation of a matrix into Jordan Canonical Form.

### Student Learning Objectives

- Target: Eigenvalues, Eigenvectors, and Eigenspaces
- Be able to show that every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue and an upper-triangular matrix with respect to some basis.
- Understand the definitions of Eigenspaces of operators; and be able show how collecting sufficiently many eigenvectors can guarantee invertibility of an operator.

- Target: The Gram-Schmidt Procedure
- Be able to apply the Gram-Schmidt procedure on any given vectors, \( v_1, \dots, v_n \), given on an inner product space \( V \), in order to generate an orthonormal basis for the subspace \( W = \mathrm{span}(v_1, \dots v_n) \) spanned by the vectors.
- Be able to use an orthonormal basis produced by,
*e.g.*the Gram-Schmidt procedure to perform orthogonal projections onto subspaces, and minimize the distance to a subspace

- Target: The Real and Complex Spectral Theorems
- Be able to apply the Real and Complex Spectral Theorems to identify operators for which diagonalization with respect to an orthonormal basis is achievable

- Target: Singular Value Decomposition
- Be able to for any linear transformation \( T \in \mathcal{L}(V,W) \) identify orthonormal bases \( \mathcal{B}(V), \) \( \mathcal{B}(W) \) consisting of the left and right singular vectors of \( T \) so that the matrix \( \mathcal{M}(T,\mathcal{B}(V), \mathcal{B}(W) ) \) is the diagonal matrix \( \Sigma = \mathrm{diag}(\sigma_1,\dots,\sigma_r) \) of singular values

- Target: Jordan Canonical Form
- Be able to, for any matrix \( A \), identify the eigenvalues, generalized eigenspaces, and Jordan Chains in order to fully identify the Jordan Basis which defines the similarity transformation into its corresponding Jordan Canonical Form

## Highly Variable Course Components

The course components described in this section are highly professor dependent, and
will probably change from semester-to-semester. They are provided as a **sample** of what the course may look like

### Required Text and Reading Materials

- Linear Algebra Done Right, 3rd Edition (2015), Sheldon Axler, Undergraduate Texts in Mathematics, Springer International Publishing
- Class web page [http://terminus.sdsu.edu/SDSU/Math524/]

### Attendance

There are no explicit penalties for missing class; however, lectures are not recorded so discussions and examples not covered in the notes will not be posted.

### Exams and Quizzes

The in-class midterms and final are designated *individual work*. A copy of the textbook in either paper or electronic form is allowed.

### Grading

- 30% Homework; approximately 10 assignments. Late assignments accepted up to 7 days after original deadline, with a 10% penalty.
- 20% Midterm 1 (In-Class)
- 10% Midterm 2 (Take-Home)
- 10% Midterm 2 (In-Class)
- 15% Final (Take-Home)
- 15% Final (In-Class)

### Schedule

Lectures are assumed to be 75 minutes in duration.

- Midterm 1 Covers
- Vector Spaces (2 lectures)
- Finite-Dimensional Vector Spaces (2 lectures)
- Linear Maps (4 lectures)

- Midterm 2 Covers (in addition)
- Polynomials (1 lecture)
- Eigenvalues, Eigenvectors, and Invariant Subspaces (3 lectures)
- Inner Product Spaces (3 lectures)

- Final (Cumulative)
- Operators on Inner Product Spaces (4 lectures)
- Applications and Computational Linear Algebra (1 lecture)
- Operators on Complex Vector Spaces (4 lectures)

## General Policies and Information

The information in this section applies to all courses offered by the department

### Students with Disabilities

If you are a student with a disability and believe you will need accommodations for this class, it is your responsibility to contact the Student Ability Success Center at (619) 594-6473. To avoid any delay in the receipt of your accommodations, you should contact Student Ability Success Center as soon as possible. Please note that accommodations are not retroactive, and that I cannot provide accommodations based upon disability until I have received an accommodation letter from Student Ability Success Center. Your cooperation is appreciated.

### Student Privacy and Intellectual Property

The Family Educational Rights and Privacy Act (FERPA) mandates the protection of student information, including contact information, grades, and graded assignments. I will not post grades or leave graded assignments in public places. Students will be notified at the time of an assignment if copies of student work will be retained beyond the end of the semester or used as examples for future students or the wider public. Students maintain intellectual property rights to work products they create as part of this course unless they are formally notified otherwise.

### Mathematics and Statistics Learning Center

The SDSU Math & Stat Learning Center is in the Love Library, Room LL-328. *"The Math and Stats Learning Center is open to support students in all lower division
math courses at SDSU. We have tutors available for walk-in help during all open hours.
TAs for Math 141, 150, 151, and 252 also hold their office hours there. Please see
the schedule of when the TAs for your class will be in the center by going to our
website: mlc.sdsu.edu. The MLC is supported by your student success fee. We strongly encourage you to use
this wonderful, free resource. Some students believe that they should not need to
ask for help. But, research has shown that the average grade for students who attend
the MLC is one half grade higher than those who don’t seek such support."*

If you are enrolled in a class which does not have targeted support, the MLC can still serve as a great math study/meeting place; and if you are interested in becoming a tutor in the center, keep an eye on the center's webpage for hiring announcements.

### Cheating and Plagiarism

Students are generally encouraged to study together, and to work together to solve
exercises. Finals, Midterms, Quizzes, Project, and other designated *"individual work"* activities must be completed without assistance. All violations will be reported
to the Center for Student Rights and Responsibilities and will also result in score/grade reductions at the professor's discretion. Please
review SDSU's full policy on academic honesty.

### Religious Observances

According to the University Policy File, students should notify the instructors of affected courses of planned absences for religious observances by the end of the second week of classes.