# Math 401 Description and Prerequisites

** Math 401,
Applications of Linear Algebra (3 credits) **
Various applications of linear algebra:
theory of finite games, linear programming,
matrix methods as applied to finite Markov chains, random walk,
incidence matrices, graphs and directed graphs,
networks and transportation problems.
The official list of topics can be found at

http://www.math.umd.edu/undergraduate/courses/syllabi/syllabusMATH401.shtml
.

However, we will substitute some of the above applications with
others.

## * Course Prerequisite *

- Math 240 or Math 461 or equivalent;

## * A Detailed Outline of Possible Topics *

** Matrices ** (Chapter 1)

Matrices and Vectors

Matrix Addition, Scalar Multiplication, and Matrix Multiplication

Transpose, Conjugate, and Hermitian Transpose

Partitioned Matrices and Matrix Operations

Diagonal, Triangular, Bidiagonal, and Tridiagonal Matrices

** Linear Algebraic Systems ** (Chapter 1)

Gaussian Elimination, Elementary Row Operations

Pivoting and Permutations

LU-Decomposition, PLU-Factorization

Matrix Inverses

** Determinants ** (Chapter 1 plus)

Alternating Multilinear Maps

Expansion by Minors, Product of Pivots

Determinants of Products

Cramer's Formula, Cofactor Matrix, Inverse Formula

** Linear Spaces ** (Chapter 2)

Vector Addition and Scalar Multiplication

Subspaces, Spans, Linear Independence

Bases and Dimension, Change of Basis

Application: Polynomial Interpolation

** Linear Mappings ** (Chapters 2 and 7)

Addition and Composition

Range and Kernel, Rank and Nullity,

Application: Directed Graphs, Incidence Matrices, and Cycles

** Euclidean Space ** (Chapter 3)

Euclidean Inner Products, Cauchy Inequality

Hermitian Positive Matricies, Hermitian Nonnegative Matricies

Principle Minor Determinants, Cholesky Factorizations

** Inner Product Spaces ** (Chapters 3 and 5)

Inner Products, Cauchy-Schwarz Inequality

Gram Matrices, Linear Independence

Application: Areas and Volumes

Orthogonal Bases, Gram-Schmidt Orthogonalization

QR-Factorization

Application: Orthogonal Polynomials

** Minimization and Least Squares ** (Chapters 4 and 5)

Orthogonal Projection Theorem

Orthogonal Subspaces, Orthogonal Decompositions

Application: Overdetermined Linear Systems

Application: Data Fitting

** Inner Products and Linear Mappings ** (Chapters 5 and 7)

Adjoints, Fredholm Alternative

Application: Sensitivity Analysis

Self-Adjoint, Skew-Adjoint, Positive Definite, Kernel Theorem

Normal, Unitary, and Isometries

Application: Graphics and Visualization

** Similar Matrices ** (Chapter 8)

Characteristic Polynomials, Cayley-Hamilton Theorem

Eigenvalues and Eigenvectors

Application: Linear Ordinary Differential Systems

Diagonalizability, Eigenspaces

Triangularizability

** Self-Adjoint and Normal Matrices ** (Chapters 7 and 8)

Schur Lemma

Spectral Decomposition for Self-Adjoint Matrices

Spectral Decomposition for Normal Matrices

Application: Discrete Fourier Series

** Rayleigh Quotients ** (Chapter 8)

Min-Max Spectral Characterization for Self-Adjoint Matrices

Application: Unconstrained Optimization Problems

Application: Constrained Optimization Problems

Singular Value Decompositions, Pseudoinverses

Application: Princple Component Analysis