A Modern Introduction to Linear Algebra (Hardcover)

Vectors

Vectors in Rⁿ

The Inner Product and Norm

Spanning Sets

Linear Independence

Bases

Subspaces

Summary

Systems of Equations

The Geometry of Systems of Equations in R² and R³

Matrices and Echelon Form

Gaussian Elimination

Computational Considerations?Pivoting

Gauss?Jordan Elimination and Reduced Row Echelon Form

Ill-Conditioned Systems of Linear Equations

Rank and Nullity of a Matrix

Systems of m Linear Equations in n Unknowns

Matrix Algebra

Addition and Subtraction of Matrices

Matrix?Vector Multiplication

The Product of Two Matrices

Partitioned Matrices

Inverses of Matrices

Elementary Matrices

The LU Factorization

Eigenvalues, Eigenvectors, and Diagonalization

Determinants

Determinants and Geometry

The Manual Calculation of Determinants

Eigenvalues and Eigenvectors

Similar Matrices and Diagonalization

Algebraic and Geometric Multiplicities of Eigenvalues

The Diagonalization of Real Symmetric Matrices

The Cayley?Hamilton Theorem (a First Look)/the Minimal Polynomial

Vector Spaces

Vector Spaces

Subspaces

Linear Independence and the Span

Bases and Dimension

Linear Transformations

Linear Transformations

The Range and Null Space of a Linear Transformation

The Algebra of Linear Transformations

Matrix Representation of a Linear Transformation

Invertible Linear Transformations

Isomorphisms

Similarity

Similarity Invariants of Operators

Inner Product Spaces

Complex Vector Spaces

Inner Products

Orthogonality and Orthonormal Bases

The Gram?Schmidt Process

Unitary Matrices and Orthogonal Matrices

Schur Factorization and the Cayley?Hamilton Theorem

The QR Factorization and Applications

Orthogonal Complements

Projections

Hermitian Matrices and Quadratic Forms

Linear Functionals and the Adjoint of an Operator

Hermitian Matrices

Normal Matrices

Quadratic Forms

Singular Value Decomposition

The Polar Decomposition

Appendix A: Basics of Set Theory

Appendix B: Summation and Product Notation

Appendix C: Mathematical Induction

Appendix D: Complex Numbers

Answers/Hints to Odd-Numbered Problems

Index

A Summary appears at the end of each chapter.