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Preface
Contents
Symbol List
The 2x2 matrices
Introduction
Definitions and operations
Some notation
Trace, transpose, and odds and ends
Determinants
Cramer's rule
Mappings
Matrices as mappings
The Cayley-Hamilton theorem
Complex numbers
M(C)
Inner products
Systems of linear equations
Introduction
Equivalent systems
Elementary row operations, echelon matrices
Solving systems of linear equations
The nxn matrices
The opening
Matrices as operators
Trace
Transpose and Hermitian adjoint
Inner product spaces
Bases of F^(n)
Change of basis of F^(n)
Invertible matrices
Matrices and bases
Bases and inner products
More on nxn matrices
Subspaces
More on subspaces
Gram-Schmidt orthogonalization process
Rank and nullity
Characteristic roots
Hermitian matrices
Triangularizing matrices with complex entries
Exponentials
Determinants
Introduction
Properties of determinants: row operations
Properties of determinants: column operations
Cramer's rule
Properties of determinants: other expansions
Elementary matrices
The determinant of the product
The characteristic polynomial
The Cayley-Hamilton theorem
Rectangular matrices More on systems of linear equations
Rectangular matrices
Linear transformations from F^(n) to F^(m)
The nullspace and column space of an mxn matrix
Abstract vector spaces
Introduction, definitions, and examples
Subspaces
Homomorphisms and isomorphisms
Isomorphisms from V to F^(n)
Linear independence in infinite-dimensional vector spaces
Inner product spaces
More on inner product spaces
Linear transformations
Introduction
Definitions, examples, and some preliminary results
Products of linear transformations
Linear transformations as matrices
Hermitian ideas
Linear transformations from one space to another
Applications
Fibonacci numbers
Equations of curves
Markov processes
Incidence models
Differential equations
Least squares methods
Approximate solutions of systems of linear equations
The approximate inverse of an mxn matrix
Solving a matrix equation using its normal equation
Finding functions that approximate data
Linear algorithms
Introduction
The LDU factorization of A
The row reduction algorithm and its inverse
Back and forward substitution Solving Ax = y
Approximate inverse and projection algorithms
A computer program for finding exact and approximate solutions
Index