Finite Dimensional Linear Algebra

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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.

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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

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

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