Linear Algebra - Ideas and Applications

Praise for the Third Edition"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications."- Electric ReviewA comprehensive introduction, Linear Algebra: Ideas and Applications, Fourth Edition provides a discussion of the theory and applications of linear algebra that blends abstract and computational concepts. With a focus on the development of mathematical intuition, the book emphasizes the need to understand both the applications of a particular technique and the mathematical ideas underlying the technique.The book introduces each new concept in the context of an explicit numerical example, which allows the abstract concepts to grow organically out of the necessity to solve specific problems. The intuitive discussions are consistently followed by rigorous statements of results and proofs.Linear Algebra: Ideas and Applications, Fourth Edition also features:* Two new and independent sections on the rapidly developing subject of wavelets* A thoroughly updated section on electrical circuit theory* Illuminating applications of linear algebra with self-study questions for additional study* End-of-chapter summaries and sections with true-false questions to aid readers with further comprehension of the presented material* Numerous computer exercises throughout using MATLAB(r) codeLinear Algebra: Ideas and Applications, Fourth Edition is an excellent undergraduate-level textbook for one or two semester courses for students majoring in mathematics, science, computer science, and engineering. With an emphasis on intuition development, the book is also an ideal self-study reference.

List of contents

PREFACE XIFEATURES OF THE TEXT XIIIACKNOWLEDGMENTS XVIIABOUT THE COMPANION WEBSITE XIX1 SYSTEMS OF LINEAR EQUATIONS 11.1 The Vector Space of m × n Matrices 1The Space Rn 4Linear Combinations and Linear Dependence 6What is a Vector Space? 11Exercises 171.1.1 Computer Projects 221.1.2 Applications to Graph Theory I 25Exercises 271.2 Systems 28Rank: The Maximum Number of Linearly Independent Equations 35Exercises 381.2.1 Computer Projects 411.2.2 Applications to Circuit Theory 41Exercises 461.3 Gaussian Elimination 47Spanning in Polynomial Spaces 58Computational Issues: Pivoting 61Exercises 63Computational Issues: Counting Flops 681.3.1 Computer Projects 691.3.2 Applications to Traffic Flow 721.4 Column Space and Nullspace 74Subspaces 77Exercises 861.4.1 Computer Projects 94Chapter Summary 952 LINEAR INDEPENDENCE AND DIMENSION 972.1 The Test for Linear Independence 97Bases for the Column Space 104Testing Functions for Independence 106Exercises 1082.1.1 Computer Projects 1132.2 Dimension 114Exercises 1232.2.1 Computer Projects 1272.2.2 Applications to Differential Equations 128Exercises 1312.3 Row Space and the rank-nullity theorem 132Bases for the Row Space 134Computational Issues: Computing Rank 142Exercises 1432.3.1 Computer Projects 146Chapter Summary 1473 LINEAR TRANSFORMATIONS 1493.1 The Linearity Properties 149Exercises 1573.1.1 Computer Projects 1623.2 Matrix Multiplication (Composition) 164Partitioned Matrices 171Computational Issues: Parallel Computing 172Exercises 1733.2.1 Computer Projects 1783.2.2 Applications to Graph Theory II 180Exercises 1813.3 Inverses 182Computational Issues: Reduction versus Inverses 188Exercises 1903.3.1 Computer Projects 1953.3.2 Applications to Economics 197Exercises 2023.4 The LU Factorization 203Exercises 2123.4.1 Computer Projects 2143.5 The Matrix of a Linear Transformation 215Coordinates 215Isomorphism 228Invertible Linear Transformations 229Exercises 2303.5.1 Computer Projects 235Chapter Summary 2364 DETERMINANTS 2384.1 Definition of the Determinant 2384.1.1 The Rest of the Proofs 246Exercises 2494.1.2 Computer Projects 2514.2 Reduction and Determinants 252Uniqueness of the Determinant 256Exercises 2584.2.1 Volume 261Exercises 2634.3 A Formula for Inverses 264Exercises 268Chapter Summary 2695 EIGENVECTORS AND EIGENVALUES 2715.1 Eigenvectors 271Exercises 2795.1.1 Computer Projects 2825.1.2 Application to Markov Processes 283Exercises 2855.2 Diagonalization 287Powers of Matrices 288Exercises 2905.2.1 Computer Projects 2925.2.2 Application to Systems of Differential Equations 293Exercises 2955.3 Complex Eigenvectors 296Complex Vector Spaces 303Exercises 3045.3.1 Computer Projects 305Chapter Summary 3066 ORTHOGONALITY 3086.1 The Scalar Product in RN 308Orthogonal/Orthonormal Bases and Coordinates 312Exercises 3166.2 Projections: The Gram-Schmidt Process 318The QR Decomposition 325Uniqueness of the QR Factorization 327Exercises 3286.2.1 Computer Projects 3316.3 Fourier Series: Scalar Product Spaces 333Exercises 3416.3.1 Application to Data Compression: Wavelets 344Exercises 3526.3.2 Computer Projects 3536.4 Orthogonal Matrices 355Householder Matrices 361Exercises 364Discrete Wavelet Transform 3676.4.1 Computer Projects 3696.5 Least Squares 370Exercises 3776.5.1 Computer Projects 3806.6 Quadratic Forms: Orthogonal Diagonalization 381The Spectral Theorem 385The Principal Axis Theorem 386Exercises 3926.6.1 Computer Projects 3956.7 The Singular Value Decomposition (SVD) 396Application of the SVD to Least-Squares Problems 402Exercises 404Computing the SVD Using Householder Matrices 406Diagonalizing Matrices Using Householder Matrices 4086.8 Hermitian Symmetric and Unitary Matrices 410Exercises 417Chapter Summary 4197 GENERALIZED EIGENVECTORS 4217.1 Generalized Eigenvectors 421Exercises 4297.2 Chain Bases 431Jordan Form 438Exercises 443The Cayley-Hamilton Theorem 445Chapter Summary 4458 NUMERICAL TECHNIQUES 4468.1 Condition Number 446Norms 446Condition Number 448Least Squares 451Exercises 4518.2 Computing Eigenvalues 452Iteration 453The QR Method 457Exercises 462Chapter Summary 464ANSWERS AND HINTS 465INDEX 487

Summary

Praise for the Third Edition This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications.

Product details

• Authors: Penney, Richard C Penney, Richard C. Penney, Richard C. (Purdue University) Penney
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 05.01.2016
• EAN: 9781118909584
• ISBN: 978-1-118-90958-4
• No. of pages: 512
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Arithmetic, algebra