Solutions Manual For Finite-Dimensional Linear Algebra, 1e by Mark Gockenbach

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