Linear algebra : ideas and applications /
Richard C . Penney, Purdue University.
- Fourth edition.
- xix, 490 pages : illustrations ; 25 cm.
Includes index.
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. 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. 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