1st Edition

Finite-Dimensional Linear Algebra

By Mark S. Gockenbach Copyright 2010
    674 Pages 50 B/W Illustrations
    by CRC Press

    Linear algebra forms the basis for much of modern mathematics—theoretical, applied, and computational. Finite-Dimensional Linear Algebra provides a solid foundation for the study of advanced mathematics and discusses applications of linear algebra to such diverse areas as combinatorics, differential equations, optimization, and approximation.

    The author begins with an overview of the essential themes of the book: linear equations, best approximation, and diagonalization. He then takes students through an axiomatic development of vector spaces, linear operators, eigenvalues, norms, and inner products. In addition to discussing the special properties of symmetric matrices, he covers the Jordan canonical form, an important theoretical tool, and the singular value decomposition, a powerful tool for computation. The final chapters present introductions to numerical linear algebra and analysis in vector spaces, including a brief introduction to functional analysis (infinite-dimensional linear algebra).

    Drawing on material from the author’s own course, this textbook gives students a strong theoretical understanding of linear algebra. It offers many illustrations of how linear algebra is used throughout mathematics.

    Some Problems Posed on Vector Spaces
    Linear equations
    Best approximation

    Fields and Vector Spaces
    Vector spaces
    Linear combinations and spanning sets
    Linear independence
    Basis and dimension
    Properties of bases
    Polynomial interpolation and the Lagrange basis
    Continuous piecewise polynomial functions

    Linear Operators
    Linear operators
    More properties of linear operators
    Isomorphic vector spaces
    Linear operator equations
    Existence and uniqueness of solutions
    The fundamental theorem; inverse operators
    Gaussian elimination
    Newton’s method
    Linear ordinary differential equations (ODEs)
    Graph theory
    Coding theory
    Linear programming

    Determinants and Eigenvalues
    The determinant function
    Further properties of the determinant function
    Practical computation of det(A)
    A note about polynomials
    Eigenvalues and the characteristic polynomial
    Eigenvalues of linear operators
    Systems of linear ODEs
    Integer programming

    The Jordan Canonical Form
    Invariant subspaces
    Generalized eigenspaces
    Nilpotent operators
    The Jordan canonical form of a matrix
    The matrix exponential
    Graphs and eigenvalues

    Orthogonality and Best Approximation
    Norms and inner products
    The adjoint of a linear operator
    Orthogonal vectors and bases
    The projection theorem
    The Gram–Schmidt process
    Orthogonal complements
    Complex inner product spaces
    More on polynomial approximation
    The energy inner product and Galerkin’s method
    Gaussian quadrature
    The Helmholtz decomposition

    The Spectral Theory of Symmetric Matrices
    The spectral theorem for symmetric matrices
    The spectral theorem for normal matrices
    Optimization and the Hessian matrix
    Lagrange multipliers
    Spectral methods for differential equations

    The Singular Value Decomposition
    Introduction to the singular value decomposition (SVD)
    The SVD for general matrices
    Solving least-squares problems using the SVD
    The SVD and linear inverse problems
    The Smith normal form of a matrix

    Matrix Factorizations and Numerical Linear Algebra
    The LU factorization
    Partial pivoting
    The Cholesky factorization
    Matrix norms
    The sensitivity of linear systems to errors
    Numerical stability
    The sensitivity of the least-squares problem
    The QR factorization
    Eigenvalues and simultaneous iteration
    The QR algorithm

    Analysis in Vector Spaces
    Analysis in Rn
    Infinite-dimensional vector spaces
    Functional analysis
    Weak convergence

    Appendix A: The Euclidean Algorithm
    Appendix B: Permutations
    Appendix C: Polynomials
    Appendix D: Summary of Analysis in R




    Mark S. Gockenbach is a professor and chair of the Department of Mathematical Sciences at Michigan Technological University.