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Linear Algebra and Linear Operators in Engineering: Volume 3 : With Applications in Mathematica (R) - H. Ted Davis

Linear Algebra and Linear Operators in Engineering: Volume 3

With Applications in Mathematica (R)

Hardcover Published: 26th June 2000
ISBN: 9780122063497
Number Of Pages: 547

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Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical sciences. Also included are several numerical applications, complete with "Mathematica" solutions and code, giving the student a "hands-on" introduction to numerical analysis. Linear Algebra and Linear Operators in Engineering is ideally suited as the main text of an introductory graduate course, and is a fine instrument for self-study or as a general reference for those applying mathematics.
- Contains numerous "Mathematica" examples complete with full code and solutions
- Provides complete numerical algorithms for solving linear and nonlinear problems
- Spans elementary notions to the functional theory of linear integral and differential equations
- Includes over 130 examples, illustrations, and exercises and over 220 problems ranging from basic concepts to challenging applications
- Presents real-life applications from chemical, mechanical, and electrical engineering and the physical sciences

Prefacep. xi
Synopsisp. 1
Matricesp. 2
Definition of a Determinantp. 3
Elementary Properties of Determinantsp. 6
Cofactor Expansionsp. 9
Cramer's Rule for Linear Equationsp. 14
Minors and Rank of Matricesp. 16
Problemsp. 18
Further Readingp. 22
Vectors and Matrices
Synopsisp. 25
Addition and Multiplicationp. 26
The Inverse Matrixp. 28
Transpose and Adjointp. 33
Partitioning Matricesp. 35
Linear Vector Spacesp. 38
Problemsp. 43
Further Readingp. 46
Solution of Linear and Nonlinear Systems
Synopsisp. 47
Simple Gauss Eliminationp. 48
Gauss Elimination with Pivotingp. 55
Computing the Inverse of a Matrixp. 58
LU-Decompositionp. 61
Band Matricesp. 66
Iterative Methods for Solving Ax = bp. 78
Nonlinear Equationsp. 85
Problemsp. 108
Further Readingp. 121
General Theory of Solvability of Linear Algebraic Equations
Synopsisp. 123
Sylvester's Theorem and the Determinants of Matrix Productsp. 124
Gauss-Jordan Transformation of a Matrixp. 129
General Solvability Theorem for Ax = bp. 133
Linear Dependence of a Vector Set and the Rank of Its Matrixp. 150
The Fredholm Alternative Theoremp. 155
Problemsp. 159
Further Readingp. 161
The Eigenproblem
Synopsisp. 163
Linear Operators in a Normed Linear Vector Spacep. 165
Basis Sets in a Normed Linear Vector Spacep. 170
Eigenvalue Analysisp. 179
Some Special Properties of Eigenvaluesp. 184
Calculation of Eigenvaluesp. 189
Problemsp. 196
Further Readingp. 203
Perfect Matrices
Synopsisp. 205
Implications of the Spectral Resolution Theoremp. 206
Diagonalization by a Similarity Transformationp. 213
Matrices with Distinct Eigenvaluesp. 219
Unitary and Orthogonal Matricesp. 220
Semidiagonalization Theoremp. 225
Self-Adjoint Matricesp. 227
Normal Matricesp. 245
Miscellaneap. 249
The Initial Value Problemp. 254
Perturbation Theoryp. 259
Problemsp. 261
Further Readingp. 278
Imperfect or Defective Matrices
Synopsisp. 279
Rank of the Characteristic Matrixp. 280
Jordan Block Diagonal Matricesp. 282
The Jordan Canonical Formp. 288
Determination of Generalized Eigenvectorsp. 294
Dyadic Form of an Imperfect Matrixp. 303
Schmidt's Normal Form of an Arbitrary Square Matrixp. 304
The Initial Value Problemp. 308
Problemsp. 310
Further Readingp. 314
Infinite-Dimensional Linear Vector Spaces
Synopsisp. 315
Infinite-Dimensional Spacesp. 316
Riemann and Lebesgue Integrationp. 319
Inner Product Spacesp. 322
Hilbert Spacesp. 324
Basis Vectorsp. 326
Linear Operatorsp. 330
Solutions to Problems Involving k-term Dyadicsp. 336
Perfect Operatorsp. 343
Problemsp. 351
Further Readingp. 353
Linear Integral Operators in a Hilbert Space
Synopsisp. 355
Solvability Theoremsp. 356
Completely Continuous and Hilbert-Schmidt Operatorsp. 366
Volterra Equationsp. 375
Spectral Theory of Integral Operatorsp. 387
Problemsp. 406
Further Readingp. 411
Linear Differential Operators in a Hilbert Space
Synopsisp. 413
The Differential Operatorp. 416
The Adjoint of a Differential Operatorp. 420
Solution to the General Inhomogeneous Problemp. 426
Green's Function: Inverse of a Differential Operatorp. 439
Spectral Theory of Differential Operatorsp. 452
Spectral Theory of Regular Sturm-Liouville Operatorsp. 459
Spectral Theory of Singular Sturm-Liouville Operatorsp. 477
Partial Differential Equationsp. 493
Problemsp. 502
Further Readingp. 509
Section 3.2: Gauss Elimination and the Solution to the Linear System Ax=bp. 511
Example 3.6.1: Mass Separation with a Staged Absorberp. 514
Section 3.7: Iterative Methods for Solving the Linear System Ax=bp. 515
Exercise 3.7.2: Iterative Solution to Ax=b--Conjugate Gradient Methodp. 518
Example 3.8.1: Convergence of the Picard and Newton-Raphson Methodsp. 519
Example 3.8.2: Steady-State Solutions for a Continuously Stirred Tank Reactorp. 521
Example 3.8.3: The Density Profile in a Liquid-Vapor Interface (Iterative Solution of an Integral Equation)p. 523
Example 3.8.4: Phase Diagram of a Polymer Solutionp. 526
Section 4.3: Gauss-Jordan Elimination and the Solution to the Linear System Ax=bp. 529
Section 5.4: Characteristic Polynomials and the Traces of a Square Matrixp. 531
Section 5.6: Iterative Method for Calculating the Eigenvalues of Tridiagonal Matricesp. 533
Example 5.6.1: Power Method for Iterative Calculation of Eigenvaluesp. 534
Example 6.2.1: Implementation of the Spectral Resolution Theorem--Matrix Functionsp. 535
Example 9.4.2: Numerical Solution of a Volterra Equation (Saturation in Porous Media)p. 537
Example 10.5.3: Numerical Green's Function Solution to a Second-Order Inhomogeneous Equationp. 540
Example 10.8.2: Series Solution to the Spherical Diffusion Equation (Carbon in a Cannonball)p. 542
Indexp. 543
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780122063497
ISBN-10: 012206349X
Series: Process Systems Engineering
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 547
Published: 26th June 2000
Country of Publication: US
Dimensions (cm): 26.26 x 18.42  x 2.82
Weight (kg): 1.15

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