MATH 4377 - Advanced Linear Algebra I & 4378 - Advanced Linear Algebra II - University of Houston

MATH 4377 - Advanced Linear Algebra I & 4378 - Advanced Linear Algebra II

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

- MATH 4377 - Advanced Linear Algebra I -

Prerequisites: MATH 2331, or equivalent, and six additional hours of 3000-4000 level Mathematics.

Course Description: Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors

Recommended Text: This syllabus is based on the Table of Contents of the textbook Linear Algebra, 5th edition, by Friedberg, Insel, Spence. ISBN: 9780134860244. The section numbers below refer to the sections in the textbook.

Math 4377 Syllabus:

1.1 Introduction (and excerpts from Appendices A (Sets), B (Functions), C (Fields), D (Complex Numbers))
1.2 Vector Spaces
1.3 Subspaces
1.4 Linear Combinations and Systems of Linear Equations
1.5 Linear Dependence and Linear Independence
1.6 Bases and Dimension
1.7* Maximal Linearly Independent Subsets
2.1 Linear Transformations, Null Spaces, and Ranges
2.2 The Matrix Represenation of a Linear Transformation
2.3 Composition of Linear Transformations and Matrix Multiplication
2.4 Invertibility and Isomorphisms
2.5 The Change of Coordinate Matrix
2.6 Dual Spaces
2.7* Homogeneous Linear Differential Equations with Constant Coefficients
3.1 Elementary Matrix Operations and Elementary Matrics
3.2 The Rank of a Matrix and Matrix Inverses
3.3 Systems of Linear Equations-Theoretical Aspects
3.4 Systems of Linear Equations-Computational Aspects
4.1 Determinants of Order 2
4.2 Determinants of Order n
4.3 Properties of Determinants
4.4 Summary -- Important Facts about Determinants
4.5* A Characterization of the Determinant
5.1 Eigenvalues and Eigenvectors (and Appendix E (Polynomials))
5.2 Diagonalizability

- 4378 - Advanced Linear Algebra II -

Prerequisites: MATH 4377.

Course Description: Similarity of matrices, diagonalization, Hermitian and positive definite matrices, normal matrices, and canonical forms, with applications

Recommended Text: This syllabus is based on the Table of Contents of the textbook Linear Algebra, 5th edition, by Friedberg, Insel, Spence. ISBN: 9780134860244. The section numbers below refer to the sections in the textbook.

Math 4378 Syllabus:

5.1 Eigenvalues and Eigenvectors (Review)
5.2 Diagonalizability (Review)
5.3* Matrix Limits and Markov Chains
5.4 Invariant Subspaces and the Cayley-Hamilton Theorem
6.1 Inner Products and Norms
6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements
6.3 The Adjoint of a Linear Operator
6.5 Unitary and Orthogonal Operators and Their Matrices
6.6 Orthogonal Projections and the Spectral Theorem
6.7* The Singular Value Decomposition and the Pseudoinverse
6.9* Einstein's Special Theory of Relativity
6.10* Conditioning and the Rayleigh Quotient
6.11* The Geometry of Orthogonal Operators
7.1 The Jordan Canonical Form I
7.2 The Jordan Canonical Form II
7.3 The Minimal Polynomial
7.4* The Rational Canonical Form

Note: The topics indicated with a * are at the discretion of the instructor. The following are some further
examples of special topics that the instructor might include if time permits:
- additional matrix theory (e.g., some of: LU decomposition, Cholesky factorization, polar decomposition, functional calculus for normal matrices, diagonal domination, eigenvalue estimates, stochastic matrices, numerical radius)
- convexity (e.g., separation, annihilator subspaces, extreme points, affine geometry)
- norms on finite dimensional vector spaces, norms of matrices
- constructions with vector spaces (e.g., the abstract (external) direct sum of vector spaces, quotient vector spaces and the isomorphism theorems, tensor products, complexifications of real vector spaces)
- vector spaces over general felds
- algebras (e.g., division algebras, Schur lemma)

CSD Accommodations:

Accommodation Forms: Students seeking academic adjustments/auxiliary aids must, in a timely manner (usually at the beginning of the semester), provide their instructor with a current Student Accommodation Form (SAF) (paper copy or online version, as appropriate) from the CSD office before an approved accommodation can be implemented.

Details of this policy, and the corresponding responsibilities of the student are outlined in The Student Academic Adjustments/Auxiliary Aids Policy (01.D.09) document under [STEP 4: Student Submission (5.4.1 & 5.4.2), Page 6]. For more information please visit the Center for Students with Disabilities Student Resources page.

Additionally, if a student is requesting a (CSD approved) testing accommodation, then the student will also complete a Request for Individualized Testing Accommodations (RITA) paper form to arrange for tests to be administered at the CSD office. CSD suggests that the student meet with their instructor during office hours and/or make an appointment to complete the RITA form to ensure confidentiality.

*Note: RITA forms must be completed at least 48 hours in advance of the original test date. Please consult your counselor ahead of time to ensure that your tests are scheduled in a timely manner. Please keep in mind that if you run over the agreed upon time limit for your exam, you will be penalized in proportion to the amount of extra time taken.

UH CAPS

Counseling and Psychological Services (CAPS) can help students who are having difficulties managing stress, adjusting to college, or feeling sad and hopeless. You can reach (CAPS) by calling 713-743-5454 during and after business hours for routine appointments or if you or someone you know is in crisis. No appointment is necessary for the "Let's Talk" program, a drop-in consultation service at convenient locations and hours around campus.