# 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**, 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)

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