Elementary Linear Algebra

by Andrilli, Stephen Francis; Hecker, David

• ISBN13: 9780120586905
• ISBN10: 0120586908
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-02-01
• Publisher: Academic Pr

Summary

Appropriate for sophomore or junior level courses, this highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study. Andrilli and Hecker give early, intensive attention to the skills necessary to make students comfortable with mathematical proof. The text builds a gradual and smooth transition from computational results to the general theory of vector spaces. It also provides flexible coverage of practical applications, exploring a comprehensive range of topics. * A unique section (1.3) is devoted exclusively to the techniques involved in reading and writing short algebraic proofs. * An illustrative example is included for most theorems. * Over 750 exercises, with over 1750 questions, range from the purely computational to the writing of proofs. * Early introduction to eigenvalues and eignvectors. * A special section (4.6) illustrates several techniques for constructing bases. * An exclusive chapter of applications offers flexible coverage.

Table of Contents

Chapter 1 Vectors and Matrices	1	(68)
Section 1.1 Fundamental Operations with Vectors	1	(15)
Section 1.2 The Dot Product	16	(11)
Section 1.3 An Introduction to Proofs	27	(18)
Section 1.4 Fundamental Operations with Matrices	45	(10)
Section 1.5 Matrix Multiplication	55	(14)
Chapter 2 Systems of Linear Equations	69	(54)
Section 2.1 Solving Systems of Linear Equations	69	(20)
Section 2.2 Equivalent Systems and Rank	89	(13)
Section 2.3 Row Space of a Matrix	102	(7)
Section 2.4 Inverses of Matrices	109	(14)
Chapter 3 Determinants and Eigenvalues	123	(50)
Section 3.1 Introduction to Determinants	123	(11)
Section 3.2 Determinants and Row Reduction	134	(10)
Section 3.3 Further Properties of the Determinant	144	(12)
Section 3.4 Eigenvalues and Diagonalization	156	(17)
Chapter 4 Finite Dimensional Vector Spaces	173	(82)
Section 4.1 Introduction to Vector Spaces	173	(11)
Section 4.2 Subspaces	184	(11)
Section 4.3 Span	195	(11)
Section 4.4 Linear Independence	206	(10)
Section 4.5 Basis and Dimension	216	(12)
Section 4.6 Constructing Special Bases	228	(11)
Section 4.7 Coordinatization	239	(16)
Chapter 5 Linear Transformations	255	(72)
Section 5.1 Introduction to Linear Transformations	255	(13)
Section 5.2 The Matrix of a Linear Transformation	268	(17)
Section 5.3 The Dimension Theorem	285	(10)
Section 5.4 Isomorphism	295	(15)
Section 5.5 Diagonalization of Linear Operators	310	(17)
Chapter 6 Orthogonality	327	(38)
Section 6.1 Orthogonal Bases and the Gram-Schmidt Process	327	(13)
Section 6.2 Orthogonal Complements	340	(13)
Section 6.3 Orthogonal Diagonalization	353	(12)
Chapter 7 Complex Vector Spaces and General Inner Products	365	(32)
Section 7.1 Complex Vector Spaces	365	(17)
Section 7.2 Inner Product Spaces	382	(15)
Chapter 8 Additional Applications	397	(78)
Section 8.1 Graph Theory	397	(8)
Section 8.2 Ohm's Law	405	(3)
Section 8.3 Least-Squares Polynomials	408	(6)
Section 8.4 Markov Chains	414	(11)
Section 8.5 Hill Substitution: An Introduction to Coding Theory	425	(4)
Section 8.6 Function Spaces	429	(3)
Section 8.7 Rotation of Axes	432	(5)
Section 8.8 Computer Graphics	437	(15)
Section 8.9 Differential Equations	452	(8)
Section 8.10 Least-Squares Solutions for Inconsistent Systems	460	(7)
Section 8.11 Quadratic Forms	467	(8)
Chapter 9 Numerical Methods	475	(28)
Section 9.1 Numerical Methods for Solving Systems	475	(13)
Section 9.2 LDU Decomposition	488	(8)
Section 9.3 The Power Method for Finding Eigenvalues	496	(7)
Appendix A Miscellaneous Proofs	503	(6)
Appendix B Functions	509	(7)
Appendix C Complex Numbers	516	(3)
Appendix D Computers and Calculators	519	(36)
Appendix E Answers to Selected Exercises	555	(29)
Index	584