Linear algebra ideas and applications

• Main Author: Penney, Richard C
• Format: eBook Book
• Language: English
• Published: Hoboken, N.J Wiley 2015; John Wiley & Sons, Incorporated; Wiley-Blackwell
• Edition: 4th ed
• Subjects: Algebras, Linear; Algebras, Linear > Textbooks; Textbooks

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 appli...

Table of Contents:

• Self-Study Questions -- Exercises -- 3.3 Inverses -- Computational Issues: Reduction versus Inverses -- Exercises -- 3.3.1 Computer Projects -- Exercises -- 3.3.2 Applications to Economics -- Self-Study Questions -- Exercises -- 3.4 The LU Factorization -- Exercises -- 3.4.1 Computer Projects -- Exercises -- 3.5 The Matrix of a Linear Transformation -- Coordinates -- Application to Differential Equations -- Isomorphism -- Invertible Linear Transformations -- Exercises -- Computer Projects -- Exercises -- Chapter Summary -- Chapter 4 Determinants -- 4.1 Definition of the Determinant -- 4.1.1 The Rest of the Proofs -- Exercises -- 4.1.2 Computer Projects -- 4.2 Reduction and Determinants -- Uniqueness of the Determinant -- Exercises -- 4.2.1 Volume -- Exercises -- A Formula for Inverses -- Exercises -- Chapter Summary -- Chapter 5 Eigenvectors and Eigenvalues -- 5.1 Eigenvectors -- Exercises -- 5.1.1 Computer Projects -- Exercises -- 5.1.2 Application to Markov Processes -- Exercises -- 5.2 Diagonalization -- Powers of Matrices -- Exercises -- 5.2.1 Computer Projects -- Exercises -- 5.2.2 Application to Systems of Differential Equations -- Exercises -- 5.3 Complex Eigenvectors -- Complex Vector Spaces -- Exercises -- 5.3.1 Computer Projects -- 5.3 Exercises -- Chapter Summary -- Chapter 6 Orthogonality -- 6.1 The Scalar Product in -- Orthogonal/Orthonormal Bases and Coordinates -- Exercises -- 6.2 Projections: The Gram-Schmidt Process -- The QR Decomposition -- Uniqueness of the Factorization -- Exercises -- 6.2.1 Computer Projects -- Exercises -- 6.3 Fourier Series: Scalar Product Spaces -- Exercises -- 6.3.1 Application to Data Compression: Wavelets -- Exercises -- 6.3.2 Computer Projects -- Exercises -- 6.4 Orthogonal Matrices -- Householder Matrices -- Exercises -- Discrete Wavelet Transform -- 6.4.1 Computer Projects -- Exercises
• Intro -- Linear Algebra -- Contents -- Preface -- Features of the Text -- Acknowledgments -- About the Companion Website -- Chapter 1 Systems of Linear Equations -- 1.1 The Vector Space of Matrices -- The Space Rn -- Linear Combinations and Linear Dependence -- What Is a Vector Space? -- Why Prove Anything? -- Exercises -- 1.1.1 Computer Projects -- Exercises -- 1.1.2 Applications to Graph Theory I -- Self-Study Questions -- Exercises -- 1.2 Systems -- Rank: The Maximum Number of Linearly Independent Equations -- Exercises -- 1.2.1 Computer Projects -- Exercises -- 1.2.2 Applications to Circuit Theory -- Self-Study Questions -- Exercises -- 1.3 Gaussian Elimination -- Spanning in Polynomial Spaces -- Computational Issues: Pivoting -- Exercises -- Computational Issues: Counting Flops -- 1.3.1 Computer Projects -- Exercises -- Applications to Traffic Flow -- Self-Study Questions -- Exercises -- 1.4 Column Space and Nullspace -- Subspaces -- Exercises -- Computer Projects -- Chapter Summary -- Chapter 2 Linear Independence and Dimension -- 2.1 The Test for Linear Independence -- Bases for the Column Space -- Testing Functions for Independence -- Exercises -- 2.1.1 Computer Projects -- Exercises -- 2.2 Dimension -- Exercises -- 2.2.1 Computer Projects -- Exercises -- 2.2.2 Applications to Differential Equations -- Exercises -- 2.3 Row Space and the rank-nullity theorem -- Bases for the Row Space -- Summary -- Computational Issues: Computing Rank -- Exercises -- 2.3.1 Computer Projects -- Exercises -- Chapter Summary -- Chapter 3 Linear Transformations -- 3.1 The Linearity Properties -- Exercises -- 3.1.1 Computer Projects -- Exercises -- 3.2 Matrix Multiplication (Composition) -- Partitioned Matrices -- Computational Issues: Parallel Computing -- Exercises -- 3.2.1 Computer Projects -- Exercises -- 3.2.2 Applications to Graph Theory II
• 6.5 Least Squares -- Exercises -- 6.5.1 Computer Projects -- Exercises -- 6.6 Quadratic Forms: Orthogonal Diagonalization -- The Spectral Theorem -- The Principal Axis Theorem -- Exercises -- 6.6.1 Computer Projects -- Exercises -- 6.7 The Singular Value Decomposition (SVD) -- Application of the SVD to Least-Squares Problems -- Exercises -- Computing the SVD Using Householder Matrices -- Diagonalizing Matrices Using Householder Matrices -- 6.8 Hermitian Symmetric and Unitary Matrices -- Exercises -- Chapter Summary -- Chapter 7 Generalized Eigenvectors -- 7.1 Generalized Eigenvectors -- Exercises -- 7.2 Chain Bases -- Jordan Form -- Exercises -- The Cayley-Hamilton Theorem -- Chapter Summary -- Chapter 8 Numerical Techniques -- 8.1 Condition Number -- Norms -- Condition Number -- Least Squares -- Exercises -- 8.2 Computing Eigenvalues -- Iteration -- The QR Method -- Proof of Theorem 8.3 on page 457 -- Exercises -- Chapter Summary -- Answers and Hints -- Section 1.1 on page 17 -- Section 1.2 on page 38 -- Section 1.2.2 on page 46 -- Section 1.3 on page 63 -- Section 1.4 on page 86 -- Section 2.1 on page 108 -- Section 2.2 on page 123 -- Section 2.2.2 on page 131 -- Section 2.3 page 143 -- Section 3.1 on page 157 -- Section 3.2 on page 173 -- Section 3.3 on page 190 -- Section 3.4 on page 212 -- Section 3.5 on page 230 -- Section 4.1 on page 249 -- Section 4.2 on page 258 -- Section 4.3 on page 268 -- Section 5.1 on page 279 -- Section 5.1.2 on page 285 -- Section 5.2 on page 290 -- Section 5.3 on page 304 -- Section 6.1 page 316 -- Section 6.2 on page 328 -- Section 6.3 on page 341 -- Section 6.4 on page 364 -- Section 6.5 on page 377 -- Section 6.6 on page 392 -- Section 6.7 on page 404 -- Section 6.8 on page 417 -- Section 7.1 on page 429 -- Section 7.2 on page 443 -- Section 8.1 on page 451 -- Index -- EULA