### Math 5101:

Advanced Linear Mathematics in Finite Dimensions

KEY COURSE TOPICS

I.   VECTOR SPACES

Why vector spaces?
Defining properties
Subspaces
Spanning sets
Linear independence
Bases and coordinates
Dimension
Linear functionals and covectors
Dual of a vector space
Bilinear functionals
Metric
Isomorphism between vector space and its dual space

II.  LINEAR TRANSFORMATIONS

Null space, range space
Dimension theorem, implicit function theorem for a linear system
Classification of linear transformations
Invertible transformations
Existence and uniqueness of a system of equations
Algebraic operations with linear transformations
The representation theorem
Change of basis, change of representation, and the transition matrix
Invariant subspaces, commuting operators and eigenvectors

III. INNER PRODUCT SPACES

Inner products
Orthogonormal bases
Gram-Schmidt orthogonalization process
Orthogonal matrices
Right and left inverses
Least squares approximation, Bessel's inequality, normal equations
The four fundamental subspaces of a matrix
The Fredholm alternative: uniqueness=existence
Intersection and sum of two vector space

IV.   EIGENVALUES AND EIGENVECTORS IN REAL VECTOR SPACES

Eigenvector basis
Diagonalizing a matrix
Generalized eigenvectors
Phase portrait of a system of linear differential equations
Powers of a matrix
Markov processes

V.   EIGENVALUES AND EIGENVECTORS IN COMPLEX VECTOR SPACES

Hermetian operators
Spectral theorem
Triangularization via unitary similarity transformation
Diagonalization of normal matrices
Positive definite matrices
Quadratic forms and the generalized eigenvalue problem
Extremization with linear constraints
Rayleigh quotient
Singular value decomposition of a rectangular matrix
Pseudo-inverse of a rectangular matrix

Text: (1) Johnson, Riess & Arnold: Introduction to Linear Algebra, 2nd, 3rd, 4th or 5th Edition
(Chapter 4 in 2nd, 3rd, or 4th Edition;  Chapter 5 in 5th Edition)
(2) Strang: Linear Algebra and its Applications, 3rd Edition
(Selected sections from Chapters 2&3, Chapter 5&6; Appendix A)

(3) Larson and Edwards: Elementary Linear Algebra, 3rd Edition
(Selected sections from Chapter 8)