# 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**

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**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

6.4 Normal and Self-Adjoint Operators

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

6.8* Bilinear and Quadratic Forms

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)

**Grading**: Please consult your instructor's syllabus regarding any and all grading guidelines.

**Justin Dart Jr. Center Accommodations:**

**Academic Adjustments/Auxiliary Aids**: The University of Houston System complies with Section 504 of the Rehabilitation Act of 1973 and the Americans with Disabilities Act of 1990, pertaining to the provision of reasonable academic adjustments/auxiliary aids for students who have a disability. In accordance with Section 504 and ADA guidelines, University of Houston strives to provide reasonable academic adjustments/auxiliary aids to students who request and require them. If you believe that you have a disability requiring an academic adjustments/auxiliary aid, please visit Justin Dart Jr. Student Accessibility Center website at https://www.uh.edu/accessibility/ for more information.

**UH CAPS**

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