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
        Adjoint of an operator

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)