Introduction to Linear Algebra

by Johnson, Lee W.; Riess, R. Dean; Arnold, Jimmy T.

• ISBN13: 9780201658590
• ISBN10: 0201658593
• Edition: 5th
• Format: Hardcover
• Copyright: 2001-07-26
• Publisher: Pearson

Summary

Introduction to Linear Algebra, 5/e is a foundation book that bridges both practical computation and theoretical principles. Due to its flexible table of contents, the book is accessible for both students majoring in the scientific, engineering, and social sciences, as well as students that want an introduction to mathematical abstraction and logical reasoning. In order to achieve the text's flexibility, the book centers on 3 principal topics: matrix theory and systems of linear equations, elementary vector space concepts, and the eigenvalue problem. This highly adaptable text can be used for a one-quarter or one-semester course at the sophomore/junior level, or for a more advanced class at the junior/senior level.

Table of Contents

Matrices and Systems of Linear Equations	1	(112)
Introduction to Matrices and Systems of Linear Equations	2	(12)
Echelon Form and Gauss--Jordan Elimination	14	(14)
Consistent Systems of Linear Equations	28	(11)
Applications (Optional)	39	(7)
Matrix Operations	46	(15)
Algebraic Properties of Matrix Operations	61	(10)
Linear Independence and Nonsingular Matrices	71	(9)
Data Fitting, Numerical Integration, and Numerical Differentiation (Optional)	80	(12)
Matrix Inverses and Their Properties	92	(21)
Vectors in 2-Space and 3-Space	113	(50)
Vectors in the Plane	114	(14)
Vectors in Space	128	(7)
The Dot Product and the Cross Product	135	(13)
Lines and Planes in Space	148	(15)
The Vector Space Rn	163	(112)
Introduction	164	(3)
Vector Space Properties of Rn	167	(9)
Examples of Subspaces	176	(12)
Bases for Subspaces	188	(14)
Dimension	202	(12)
Orthogonal Bases for Subspaces	214	(11)
Linear Transformations from Rn to Rm	225	(18)
Least-Squares Solutions to Inconsistent Systems, with Applications to Data Fitting	243	(12)
Theory and Practice of Least Squares	255	(20)
The Eigenvalue Problem	275	(82)
The Eigenvalue Problem for (2 × 2) Matrices	276	(4)
Determinants and the Eigenvalue Problem	280	(10)
Elementary Operations and Determinants (Optional)	290	(8)
Eigenvalues and the Characteristic Polynomial	298	(9)
Eigenvectors and Eigenspaces	307	(8)
Complex Eigenvalues and Eigenvectors	315	(10)
Similarity Transformations and Diagonalization	325	(13)
Difference Equations; Markov Chains; Systems of Differential Equations (Optional)	338	(19)
Vector Spaces and Linear Transformations	357	(90)
Introduction	358	(2)
Vector Spaces	360	(8)
Subspaces	368	(7)
Linear Independence, Bases, and Coordinates	375	(13)
Dimension	388	(4)
Inner-Product Spaces, Orthogonal Bases, and Projections (Optional)	392	(11)
Linear Transformations	403	(8)
Operations with Linear Transformations	411	(8)
Matrix Representations for Linear Transformations	419	(12)
Change of Basis and Diagonalization	431	(16)
Determinants	447	(36)
Introduction	448	(1)
Cofactor Expansions of Determinants	448	(7)
Elementary Operations and Determinants	455	(10)
Cramer's Rule	465	(6)
Applications of Determinants: Inverses and Wronksians	471	(12)
Eigenvalues and Applications	483
Quadratic Forms	484
Systems of Differential Equations	493
Transformation to Hessenberg Form	502
Eigenvalues of Hessenberg Matrices	510
Householder Transformations	519
The QR Factorization and Least-Squares Solutions	531
Matrix Polynomials and the Cayley--Hamilton Theorem	540
Generalized Eigenvectors and Solutions of Systems of Differential Equations	546
Appendix: An Introduction to MATLAB	AP1
Answers to Selected Odd-Numbered Exercises	AN1
Index	I1