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
Diagonalization
Summary
Fields and Vector Spaces
Fields
Vector spaces
Subspaces
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
Diagonalization
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
Bibliography
Index
Biography
Mark S. Gockenbach is a professor and chair of the Department of Mathematical Sciences at Michigan Technological University.
