9780131473829

Applied Linear Algebra

by ;
  • ISBN13:

    9780131473829

  • ISBN10:

    0131473824

  • Edition: 1st
  • Format: Paperback
  • Copyright: 1/10/2005
  • Publisher: Pearson

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Summary

This book describes basic methods and algorithms used in modern, real problems likely to be encountered by engineers and scientists - and fosters an understanding of why mathematical techniques work and how they can be derived from first principles.Assumes no previous exposure to linear algebra. Presents applications hand in hand with theory, leading readers through the reasoning that leads to the important results. Provides theorems and proofs where needed. Features abundant exercises after almost every subsection, in a wide range of difficulty.A thorough reference for engineers and scientists.

Table of Contents

Preface xi
Linear Algebraic Systems
1(76)
Solution of Linear Systems
2(2)
Matrices and Vectors
4(8)
Matrix Arithmetic
6(6)
Gaussian Elimination---Regular Case
12(10)
Elementary Matrices
16(2)
The L U Factorization
18(2)
Forward and Back Substitution
20(2)
Pivoting and Permutations
22(9)
Permutation Matrices
25(2)
The Permuted L U Factorization
27(4)
Matrix Inverses
31(13)
Gauss--Jordan Elimination
36(5)
Solving Linear Systems with the Inverse
41(1)
The L D V Factorization
42(2)
Transposes and Symmetric Matrices
44(5)
Factorization of Symmetric Matrices
46(3)
Practical Linear Algebra
49(11)
Tridiagonal Matrices
53(3)
Pivoting Strategies
56(4)
General Linear Systems
60(10)
Homogeneous Systems
68(2)
Determinants
70(7)
Vector Spaces and Bases
77(53)
Real Vector Spaces
78(5)
Subspaces
83(6)
Span and Linear Independence
89(11)
Linear Independence and Dependence
94(6)
Bases and Dimension
100(7)
The Fundamental Matrix Subspaces
107(15)
Kernel and Range
107(4)
The Superposition Principle
111(3)
Adjoint Systems, Cokernel, and Corange
114(2)
The Fundamental Theorem of Linear Algebra
116(6)
Graphs and Incidence Matrices
122(8)
Inner Products and Norms
130(50)
Inner Products
131(6)
Inner Products on Function Spaces
134(3)
Inequalities
137(7)
The Cauchy--Schwarz Inequality
138(2)
Orthogonal Vectors
140(2)
The Triangle Inequality
142(2)
Norms
144(9)
Unit Vectors
147(3)
Equivalence of Norms
150(3)
Positive Definite Matrices
153(10)
Gram Matrices
157(6)
Completing the Square
163(6)
The Cholesky Factorization
168(1)
Complex Vector Spaces
169(11)
Complex Numbers
170(4)
Complex Vector Spaces and Inner Products
174(6)
Minimization and Least Squares Approximation
180(37)
Minimization Problems
180(4)
Equilibrium Mechanics
181(1)
Solution of Equations
181(2)
The Closest Point
183(1)
Minimization of Quadratic Functions
184(5)
Least Squares and the Closest Point
189(7)
Least Squares
193(3)
Data Fitting and Interpolation
196(21)
Polynomial Approximation and Interpolation
201(8)
Approximation and Interpolation by General Functions
209(2)
Weighted Least Squares
211(2)
Least Squares Approximation in Function Spaces
213(4)
Orthogonality
217(76)
Orthogonal Bases
218(9)
Computations in Orthogonal Bases
222(5)
The Gram--Schmidt Process
227(8)
Modifications of the Gram--Schmidt Process
232(3)
Orthogonal Matrices
235(13)
The Q R Factorization
240(3)
III-Conditioned Systems and Householder's Method
243(5)
Orthogonal Polynomials
248(8)
The Legendre Polynomials
249(4)
Other Systems of Orthogonal Polynomials
253(3)
Orthogonal Projections and Least Squares
256(12)
Orthogonal Projection
256(4)
Orthogonal Least Squares
260(5)
Orthogonal Polynomials and Least Squares
265(3)
Orthogonal Subspaces
268(9)
Orthogonality of the Fundamental Matrix Subspaces and the Fredholm Alternative
272(5)
Discrete Fourier Series and the Fast Fourier Transform
277(16)
Compression and Noise Removal
285(2)
The Fast Fourier Transform
287(6)
Equilibrium
293(37)
Springs and Masses
294(7)
Positive Definiteness and the Minimization Principle
299(2)
Electrical Networks
301(11)
Batteries, Power, and the Electrical--Mechanical Correspondence
307(5)
Structures
312(18)
Linearity
330(60)
Linear Functions
331(15)
Linear Operators
336(2)
The Space of Linear Functions
338(3)
Composition
341(3)
Inverses
344(2)
Linear Transformations
346(12)
Change of Basis
353(5)
Affine Transformations and Isometries
358(5)
Isometry
360(3)
Linear Systems
363(19)
The Superposition Principle
365(5)
Inhomogeneous Systems
370(5)
Superposition Principles for Inhomogeneous Systems
375(2)
Complex Solutions to Real Systems
377(5)
Adjoints
382(8)
Self--Adjoint and Positive Definite Linear Functions
385(2)
Minimization
387(3)
Eigenvalues
390(55)
Simple Dynamical Systems
391(4)
Scalar Ordinary Differential Equations
391(3)
First Order Dynamical Systems
394(1)
Eigenvalues and Eigenvectors
395(11)
Basic Properties of Eigenvalues
401(5)
Eigenvector Bases and Diagonalization
406(7)
Diagonalization
409(4)
Eigenvalues of Symmetric Matrices
413(12)
The Spectral Theorem
418(4)
Optimization Principles for Eigenvalues
422(3)
Singular Values
425(9)
Condition Number, Rank, and Principal Component Analysis
429(1)
The Pseudoinverse
430(4)
Incomplete Matrices
434(11)
The Schur Decomposition
435(3)
The Jordan Canonical Form
438(7)
Linear Dynamical Systems
445(65)
Basic Solution Techniques
446(13)
The Phase Plane
447(3)
Existence and Uniqueness
450(2)
Complete Systems
452(4)
The General Case
456(3)
Stability of Linear Systems
459(6)
Two-Dimensional Systems
465(7)
Distinct Real Eigenvalues
466(2)
Complex Conjugate Eigenvalues
468(1)
Incomplete Double Real Eigenvalue
469(1)
Complete Double Real Eigenvalue
469(3)
Matrix Exponentials
472(13)
Inhomogeneous Linear Systems
478(3)
Applications in Geometry
481(4)
Dynamics of Structures
485(15)
Stable Structures
487(5)
Unstable Structures
492(3)
Systems with Differing Masses
495(2)
Friction and Damping
497(3)
Forcing and Resonance
500(10)
Electrical Circuits
504(2)
Forcing and Resonance in Systems
506(4)
Iteration of Linear Systems
510(73)
Linear Iterative Systems
511(12)
Scalar Systems
511(3)
Powers of Matrices
514(6)
Diagonalization and Iteration
520(3)
Stability
523(7)
Fixed Points
528(2)
Matrix Norms and the Gerschgorin Theorem
530(10)
Matrix Norms
530(2)
Explicit Formulae
532(4)
The Gerschgorin Circle Theorem
536(4)
Markov Processes
540(6)
Iterative Solution of Linear Algebraic Systems
546(22)
The Jacobi Method
549(3)
The Gauss--Seidel Method
552(5)
Successive Over-Relaxation (SOR)
557(5)
Conjugate Gradients
562(6)
Numerical Computation of Eigenvalues
568(15)
The Power Method
568(4)
The Q R Algorithm
572(6)
Tridiagonalization
578(5)
Boundary Value Problems in One Dimension
583(67)
Elastic Bars
584(9)
Generalized Functions and the Green's Function
593(19)
The Delta Function
594(4)
Calculus of Generalized Functions
598(8)
The Green's Function
606(6)
Adjoints and Minimum Principles
612(11)
Adjoints of Differential Operators
612(4)
Positivity and Minimum Principles
616(4)
Inhomogeneous Boundary Conditions
620(3)
Beams and Splines
623(12)
Splines
630(5)
Sturm--Liouville Boundary Value Problems
635(5)
Finite Elements
640(10)
References 650(3)
Answers to Selected Exercises 653(46)
Index 699

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