Linear Algebra and Its Applications

by Lay, David C.

• ISBN13: 9780201347746
• ISBN10: 0201347741
• Additional ISBN(s): 9780201709704, 9780201824780
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Addison-Wesley
• More Book Details
• Purchase Benefits

Summary

Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text. Finally, when discussed in the abstract, these concepts are more accessible. Students' conceptual understanding is reinforced through True/False questions, practice problems, and the use of technology. David Lay changed the face of linear algebra with the execution of this philosophy, and continues his quest to improve the way linear algebra is taught with the new Updated Second Edition. With this update, he builds on this philosophy through increased visualization in the text, vastly enhanced technology support, and an extensive instructor support package. He has added additional figures to the text to help students visualize abstract concepts at key points in the course. A new dedicated CD and Website further enhance the course materials by providing additional support to help students gain command of difficult concepts. The CD, included in the back of the book, contains a wealth of new materials, with a registration coupon allowing access to a password-protected Website. These new materials are tied directly to the text, providing a comprehensive package for teaching and learning linear algebra.

Table of Contents

Preface	xiii
A Note To Students	xix
Linear Equations in Linear Algebra	1	(96)
Introductory Example: Linear Models in Economics and Engineering	1	(96)
Systems of Linear Equations	2	(11)
Row Reduction and Echelon Forms	13	(14)
Vector Equations	27	(12)
The Matrix Equation Ax = b	39	(9)
Solution Sets of Linear Systems	48	(10)
Linear Independence	58	(8)
Introduction to Linear Transformations	66	(10)
The Matrix of a Linear Transformation	76	(9)
Linear Models in Business, Science, and Engineering	85	(10)
Supplementary Exercises	95	(2)
Matrix Algebra	97	(82)
Introductory Example: Computer Graphics in Automotive Design	97	(82)
Matrix Operations	98	(12)
The Inverse of a Matrix	110	(10)
Characterizations of Invertible Matrices	120	(5)
Partitioned Matrices	125	(8)
Matrix Factorizations	133	(10)
Iterative Solutions of Linear Systems	143	(5)
The Leontief Input-Output Model	148	(7)
Applications to Computer Graphics	155	(10)
Subspaces of Rn	165	(12)
Supplementary Exercises	177	(2)
Determinants	179	(30)
Introductory Example: Determinants in Analytic Geometry	179	(30)
Introduction to Determinants	180	(7)
Properties of Determinants	187	(8)
Cramer's Rule, Volume, and Linear Transformations	195	(11)
Supplementary Exercises	206	(3)
Vector Spaces	209	(86)
Introductory Example: Space Flight and Control Systems	209	(86)
Vector Spaces and Subspaces	210	(10)
Null Spaces, Column Spaces, and Linear Transformations	220	(11)
Linearly Independent Sets; Bases	231	(9)
Coordinate Systems	240	(10)
The Dimension of a vector Space	250	(7)
Rank	257	(8)
Change of Basis	265	(6)
Applications to Difference Equations	271	(11)
Applications to Markov Chains	282	(10)
Supplementary Exercises	292	(3)
Eigenvalues and Eigenvectors	295	(72)
Introductory Example: Dynamical Systems and Spotted Owls	295	(72)
Eigenvectors and Eigenvalues	296	(9)
The Characteristic Equation	305	(8)
Diagonalization	313	(8)
Eigenvectors and Linear Transformations	321	(8)
Complex Eigenvalues	329	(7)
Discrete Dynamical Systems	336	(11)
Applications to Differential Equations	347	(10)
Iterative Estimates for Eigenvalues	357	(8)
Supplementary Exercises	365	(2)
Orthogonality and Least-Squares	367	(74)
Introductory Example: Readjusting the North American Datum	367	(74)
Inner Product, Length, and Orthogonality	369	(9)
Orthogonal Sets	378	(11)
Orthogonal Projections	389	(8)
The Gram-Schmidt Process	397	(7)
Least-Squares Problems	404	(10)
Applications to Linear Models	414	(8)
Inner Product Spaces	422	(9)
Applications of Inner Product Spaces	431	(8)
Supplementary Exercises	439	(2)
Symmetric Matrices and Quadratic Forms	441
Introductory Example: Multichannel Image Processing	441
Diagonalization of Symmetric Matrices	443
Quadratic Forms	450
Constrained Optimization	458
The Singular Value Decomposition	466
Applications to Image Processing and Statistics	477
Supplementary Exercises	485
Appendixes
A Uniqueness of the Reduced Echelon Form	A1
B Complex Numbers	A3
Glossary	A9
Answers to Odd-Numbered Exercises	A21
Index	I1