Matrices and Linear Transformations: Second Edition(Dover Books on Mathema 1.4tics)

Undergraduate-level introduction to linear algebra and matrix theory. Explores matrices and linear systems, vector spaces, determinants, spectral decomposition, Jordan canonical form, much more. "Comprehensive . . . an excellent introduction to the subject." - Electronic Engineer's Design Magazine. This introductory textbook, aimed at sophomore- and junior-level undergraduates in mathematics, engineering, and the physical sciences, offers a smooth, in-depth treatment of linear algebra and matrix theory. The major objects of study are matrices over an arbitrary field. Contents include Matrices and Linear Systems; Vector Spaces; Determinants; Linear Transformations; Similarity: Part I and Part II; Polynomials and Polynomial Matrices; Matrix Analysis; and Numerical Methods. The first seven chapters, which require only a first course in calculus and analytic geometry, deal with matrices and linear systems, vector spaces, determinants, linear transformations, similarity, polynomials, and polynomial matrices. Chapters 8 and 9, parts of which require the student to have completed the normal course sequence in calculus and differential equations, provide introductions to matrix analysis and numerical linear algebra, respectively. Among the key features are coverage of spectral decomposition, the Jordan canonical form, the solution of the matrix equation AX = XB, and over 375 problems, many with answers.

Table of Contents:
Chapter I Matrices and Linear Systems 1.1 Introduction 1.2 Fields and number systems 1.3 Matrices 1.4 Matrix addition and scalar multiplication 1.5 Transposition 1.6 Partitioned matrices 1.7 Special kinds of matrices 1.8 Row equivalence 1.9 Elementary matrices and matrix Inverses 1.10 Column equivalence 1.11 Equivalence Chapter 2 Vector Spaces 2.1 Introduction 2.2 Subspaces 2.3 Linear independence and bases 2.4 The rank of a matrix 2.5 Coordinates and isomorphisms 2.6 Uniqueness theorem for row equivalence. Chapter 3 Determinants 3.1 Definition of the determinant 3.2 The Laplace expansion 3.3 Adjoints and inverses 3.4 Determinants and rank Chapter 4 Linear Transformations 4.1 Definition and examples 4.2 Matrix representation 4.3 Products and inverses 4.4 Change of basis and similarity 4.5 Characteristic vectors and characteristic values 4.6 Orthogonality and length 4.7 Gram-Schmidt process 4.8 Schur's theorem and normal matrices Chapter 5 Similarity: Part I 5.1 The Cayley-Hamilton theorem 5.2 Direct sums and invariant subspaces 5.3 Nilpotent linear operators 5.4 The Jordan canonical form 5.5 Jordan form-continued 5.6 Commutativity (the equation AX = XB) Chapter 6 Polynomials and Polynomial Matrices 6.1 Introduction and review 6.2 Divisibility and irreducibility 6.3 Lagrange interpolation 6.4 Matrices with polynomial elements 6.5 Equivalence over F[x] . 6.6 Equivalence and similarity Chapter 7 Similarity: Part II 7.1 Nonderogatory matrices 7.2 Elementary divisors 7.3 The classical canonical form 7.4 Spectral decomposition 7.5 Polar decomposition Chapter 8 Matrix Analysis 8.1 Sequences and series 8.2 Primary functions 8.3 Matrices of functions 8.4 Systems of linear differential equations Chapter 9 Numerical Methods 9.1 Introduction 9.2 Exact methods for solving AX = K 9.3 Iterative methods for solving AX = K 9.4 Characteristic values and vectors Answers to Selected Exercises Appendix Glossary of Mathematical Symbols Index