Math 401 Description and Prerequisites

UM Undergraduate Catalog Course Description

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