Linear Algebra and Its Applications with CD-ROM, Update

by Lay, David C.

• ISBN13: 9780321287137
• ISBN10: 0321287134
• Edition: 3rd
• Format: Hardcover w/CD
• Copyright: 2006-01-01
• Publisher: Addison Wesley

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 so that when discussed in the abstract, these concepts are more accessible.

Table of Contents

Preface	ix
A Note to Students	xv
Linear Equations in Linear Algebra	1	(104)
Introductory Example: Linear Models in Economics and Engineering	1	(104)
Systems of Linear Equations	2	(12)
Row Reduction and Echelon Forms	14	(14)
Vector Equations	28	(12)
The Matrix Equation Ax = b	40	(10)
Solution Sets of Linear Systems	50	(7)
Applications of Linear Systems	57	(8)
Linear Independence	65	(8)
Introduction to Linear Transformations	73	(9)
The Matrix of a Linear Transformation	82	(10)
Linear Models in Business, Science, and Engineering	92	(10)
Supplementary Exercises	102	(3)
Matrix Algebra	105	(80)
Introductory Example: Computer Models in Aircraft Design	105	(80)
Matrix Operations	107	(11)
The Inverse of a Matrix	118	(10)
Characterizations of Invertible Matrices	128	(6)
Partitioned Matrices	134	(8)
Matrix Factorizations	142	(10)
The Leontief Input-Output Model	152	(6)
Applications to Computer Graphics	158	(9)
Subspaces of Rn	167	(9)
Dimension and Rank	176	(7)
Supplementary Exercises	183	(2)
Determinants	185	(30)
Introductory Example: Determinants in Analytic Geometry	185	(30)
Introduction to Determinants	186	(6)
Properties of Determinants	192	(9)
Cramer's Rule, Volume, and Linear Transformations	201	(10)
Supplementary Exercises	211	(4)
Vector Spaces	215	(86)
Introductory Example: Space Flight and Control Systems	215	(86)
Vector Spaces and Subspaces	216	(10)
Null Spaces, Column Spaces, and Linear Transformations	226	(11)
Linearly Independent Sets: Bases	237	(9)
Coordinate Systems	246	(10)
The Dimension of a Vector Space	256	(6)
Rank	262	(9)
Change of Basis	271	(6)
Applications to Difference Equations	277	(11)
Applications to Markov Chains	288	(10)
Supplementary Exercises	298	(3)
Eigenvalues and Eigenvectors	301	(72)
Introductory Example: Dynamical Systems and Spotted Owls	301	(72)
Eigenvectors and Eigenvalues	302	(8)
The Characteristic Equation	310	(9)
Diagonalization	319	(8)
Eigenvectors and Linear Transformations	327	(8)
Complex Eigenvalues	335	(7)
Discrete Dynamical Systems	342	(11)
Applications to Differential Equations	353	(10)
Iterative Estimates for Eigenvalues	363	(7)
Supplementary Exercises	370	(3)
Orthogonality and Least Squares	373	(74)
Introductory Example: Readjusting the North American Datum	373	(74)
Inner Product, Length, and Orthogonality	375	(9)
Orthogonal Sets	384	(10)
Orthogonal Projections	394	(8)
The Gram--Schmidt Process	402	(7)
Least-Squares Problems	409	(10)
Applications to Linear Models	419	(8)
Inner Product Spaces	427	(9)
Applications of Inner Product Spaces	436	(8)
Supplementary Exercises	444	(3)
Symmetric Matrices and Quadratic Forms	447
Introductory Example: Multichannel Image Processing	447
Diagonalization of Symmetric Matrices	449	(6)
Quadratic Forms	455	(8)
Constrained Optimization	463	(8)
The Singular Value Decomposition	471	(11)
Applications to Image Processing and Statistics	482	(9)
Supplementary Exercises	491
The Geometry of Vector Spaces
Introductory Example: The Platonic Solids
Affine Combinations
Affine Independence
Convex Combinations
Hyperplanes
Polytopes
Curves and Surfaces
Optimization
Introductory Example: The Berlin Airlift
Matrix Games
Linear Programming---Geometric Method
Linear Programming---Simplex Method
Duality
Appendixes
A Uniqueness of the Reduced Echelon Form	1	(2)
B Complex Numbers	3	(6)
Glossary	9	(10)
Answers to Odd-Numbered Exercises	19
Index	1