Math 524 — Linear Algebra — Syllabus

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

1. Math 245 (Discrete Mathematics), and
2. One of:
   1. Math 254 (Introduction to Linear Algebra), or
   2. Math 342A (Methods of Applied Mathematics), or
   3. 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.

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.

• 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

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.

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.

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.

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.

Examples of academic dishonesty include but are not limited to

• copying, in part or in whole, from another's test or other examination
• obtaining copies of a test, an examination, or other course material without the permission of the instructor
• collaborating with another or others in work to be presented without the permission of the instructor
• falsifying records, laboratory work, or other course data
• submitting work previously presented in another course, if contrary to the rules of the course
• altering or interfering with grading procedures
• assisting another student in any of the above
• using sources verbatim or paraphrasing without giving proper attribution (this can include phrases, sentences, paragraphs and/or pages of work)
• copying and pasting work from an online or offline source directly and calling it your own
• using information you find from an online or offline source without giving the author credit
• replacing words or phrases from another source and inserting your own words or phrases

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Students are instructed to contact their professor/instructor in the event they need to miss class, etc. due to an illness, injury or emergency. All decisions about the impact of an absence, as well as any arrangements for making up work, rest with the instructors. Student Health Services (SHS) does not provide medical excuses for short-term absences due to illness or injury. When a medical-related absence persists beyond five days, SHS will work with students to provide appropriate documentation. When a student is hospitalized or has a serious, ongoing illness or injury, SHS will, at the student's request and with the student’s consent, communicate with the student’s instructors via the Vice President for Student Affairs and may communicate with the student’s Assistant Dean and/or the Student Ability Success Center

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