Linear Algebra With Applications

by Williams, Gareth

• ISBN13: 9780763714512
• ISBN10: 0763714518
• Edition: 4th
• Format: Hardcover
• Copyright: 2000-07-01
• Publisher: Jones & Bartlett Learning

Table of Contents

Preface	v
Systems of Linear Equations	1	(42)
Matrices and Systems of Linear Equations	2	(14)
Many Systems	11	(5)
Gauss-Jordan Elimination	16	(12)
Homogeneous Systems of Linear Equations	23	(5)
Curve Fitting, Electrical Networks, and Traffic Flow	28	(15)
Curve Fitting	28	(2)
Electrical Network Analysis	30	(4)
Traffic Flow	34	(9)
Matrices	43	(76)
Addition, Scalar Multiplication, and Multiplication of Matrices	44	(13)
Addition of Matrices	45	(1)
Scalar Multiplication of Matrices	45	(1)
Negation and Subtraction	46	(1)
Multiplication of Matrices	47	(3)
Size of a Product Matrix	50	(2)
Matrix Notation and Systems of Equations	52	(1)
Fast Matrix Multiplication	52	(5)
Properties of Matrix Operations	57	(8)
Powers of Matrices	62	(3)
Symmetric Matrices and Seriation in Archaeology	65	(13)
Matrices with Complex Elements	70	(2)
Seriation in Archaeology	72	(6)
The Inverse of a Matrix and Cryptography	78	(11)
Notation	79	(1)
Gauss-Jordan Elimination for finding the Inverse of a Matrix	80	(5)
Cryptography	85	(4)
The Leontief Input-Output Model in Economics	89	(6)
Markov Chains, Population Movements, and Genetics	95	(9)
A Communication Model and Group Relationships in Sociology	104	(15)
Distance in a Digraph	107	(1)
Group Relationships in Sociology	108	(11)
Determinants	119	(34)
Introduction to Determinants	120	(9)
Computing Determinants of 2 X 2 and 3 X 3 Matrices	124	(1)
Alternative Definition of Determinant	125	(4)
Properties of Determinants	129	(7)
Numerical Evaluation of a Determinant	136	(5)
Numerical Evaluation of a Determinant	137	(4)
Determinants, Matrix Inverses, and Systems of Linear Equations	141	(12)
The Vector Space Rn	153	(54)
Introduction to Vectors	154	(10)
Zero Vector	158	(1)
Negative Vector	159	(1)
Subtraction	159	(3)
Column Vectors	162	(2)
Dot Product, Norm, Angle, and Distance	164	(14)
Norm of a Vector in Rn	165	(4)
Angle between Vectors	169	(4)
Distance between Points	173	(5)
Introduction to Linear Transformations	178	(7)
Composition of Matrix Transformations	182	(3)
Matrix Transformations, Computer Graphics, and Fractals	185	(22)
Rotation about the Origin	185	(2)
Matrix Representation	187	(1)
Dilation and Contraction	188	(2)
Reflection	190	(1)
Transformations Defined by Nonsingular Matrices	191	(2)
Translations and Affine Transformations	193	(3)
Transformations in Computer Graphics	196	(2)
Fractal Pictures of Nature	198	(9)
General Vector Spaces	207	(68)
Vector Spaces	208	(5)
Vector Spaces of Matrices	209	(1)
Vector Spaces of Functions	210	(1)
The Complex Vector Space Cn	211	(2)
Subspaces	213	(7)
Linear Combinations of Vectors	220	(11)
Linear Dependence and Independence	231	(7)
Linear Dependence of {v1, v2}	234	(1)
Linear Dependence of {v1, v2, v3}	235	(3)
Bases and Dimension	238	(10)
Rank of a Matrix	248	(9)
Orthonormal Vectors and Projections in Rn	257	(18)
Projection of One Vector onto Another Vector	260	(4)
Projection of a Vector onto a Subspace	264	(3)
Distance of a Point from a Subspace	267	(2)
Orthogonal Matrices	269	(6)
Eigenvalues and Eigenvectors	275	(42)
Eigenvalues and Eigenvectors	276	(7)
Computation of Eigenvalues and Eigenvectors	276	(7)
Demography and Weather Prediction	283	(6)
Diagonalization of Matrices	289	(11)
Diagonalization of Symmetric Matrices	295	(1)
Orthogonal Diagonalization	296	(4)
Quadratic Forms, Difference Equations, and Normal Modes	300	(17)
Rotation of Coordinates	300	(2)
Quadratic Forms	302	(3)
Difference Equations	305	(3)
Fibonacci Sequencel	308	(1)
Normal Modes of Oscillating Systems	309	(8)
Linear Transformations	317	(50)
Linear Transformations, Kernel, and Range	318	(15)
Terminology	327	(6)
Transformations and Systems of Linear Equations	333	(8)
Homogeneous Equations	334	(1)
Nonhomogeneous Equations	335	(3)
Many Systems	338	(3)
Coordinate Vectors	341	(10)
Notation	342	(3)
Change of Basis	345	(3)
Isomorphisms	348	(3)
Matrix Representations of Linear Transformations	351	(16)
Importance of Matrix Representations	355	(3)
Relations between Matrix Representations	358	(2)
Diagonal Matrix Representation of a Linear Operator	360	(7)
Inner Product Spaces	367	(42)
Inner Product Spaces	368	(10)
Norm of a Vector	371	(1)
Angle	372	(1)
Orthogonal Vectors	373	(1)
Distance	373	(1)
Inner Product on Cn	374	(4)
Non-Euclidean Geometry and Special Relativity	378	(7)
Special Relativity	380	(5)
Approximation of Functions and Coding Theory	385	(7)
Fourier Approximations	388	(2)
Coding Theory	390	(2)
Least-Squares Curves	392	(17)
Least-Squares Curves	395	(14)
Numerical Techniques	409	(44)
Gaussian Elimination	410	(6)
Comparison of Gauss-Jordan and Gaussian Elimination	413	(3)
The Method of LU Decomposition	416	(8)
Method of LU Decomposition	418	(3)
Construction of an LU decomposition of A	421	(3)
Practical Difficulties in Solving Systems of Equations	424	(12)
The Condition Number of a Matrix	424	(5)
Pivoting and Scaling Techniques	429	(3)
Pivoting and Scaling Procedures	432	(4)
Iterative Methods for Solving Systems of Linear Equations	436	(5)
Jacobi Method1	436	(2)
Gauss-Seidel Method2	438	(2)
Comparison of Gaussian Elimination and Gauss-Seidel	440	(1)
Eigenvalues by Iteration; Connectivity of Networks	441	(12)
Deflation	446	(1)
Accessibility Index of a Network	446	(7)
Linear Programming	453	(26)
A Geometrical Introduction to Linear Programming	454	(10)
A Linear Programming Problem	456	(3)
Minimum Value of a Function	459	(2)
Discussion of the Method	461	(3)
The Simplex Method	464	(7)
Geometrical Explanation of the Simplex Method	471	(8)
Appendix A Cross Product	479	(12)
Appendix B Equations of Planes and Lines in Three-Space	491	(8)
Appendix C Graphing Calculator Manual	499	(14)
Appendix D MATLAB Manual	513	(62)
Answers to Selected Exercises	575	(64)
Index	639