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 2318, and MATH 3325 and three 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)