A Modern Introduction to Linear Algebra

Henry Ricardo



A Modern Introduction to Linear Algebra provides a rigorous yet accessible matrix-oriented introduction to the essential concepts of linear algebra. Concrete, easy-to-understand examples motivate the theory.

The book first discusses vectors, Gaussian elimination, and reduced row echelon forms. It then offers a thorough introduction to matrix algebra, including defining the determinant naturally from the PA=LU factorization of a matrix. The author goes on to cover finite-dimensional real vector spaces, infinite-dimensional spaces, linear transformations, and complex vector spaces. The final chapter presents Hermitian and normal matrices as well as quadratic forms.

Taking a computational, algebraic, and geometric approach to the subject, this book provides the foundation for later courses in higher mathematics. It also shows how linear algebra can be used in various areas of application. Although written in a "pencil and paper" manner, the text offers ample opportunities to enhance learning with calculators or computer usage.

Features

• Develops key concepts in the context of Euclidean n-space
• Explains the theory of matrices
• Explores the differences between finite- and infinite-dimensional vector spaces
• Covers the algebra and matrix representation of linear transformations
• Presents proofs for nearly all results
• Includes a host of examples and various applications reflecting some of the many disciplines that use linear algebra

Table of Contents

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