Linear algebra: ideas and applications

Linear Algebra: Ideas and Applications, Third Edition has been updated and revised, but its parallel structure (abstract concepts are introduced along withcomputational) and concepts remain intact. The book covers a number of applications of linear algebra and features a unique treatment of vector spaces, proofs, and computations. The text explores linear algebra using an approach thatintroduces abstract concepts as they are needed to fully understand the computations. INDICE: Preface. Features of the Text. 1. Systems of Linear Equations. 1.1The Vector Space of m x n Matrices. The Space Rn. Linear Combinations and Linear Dependence. What Is a Vector Space? Why Prove Anything? True-False Questions. Exercises. 1.1.1 Computer Projects. Exercises. 1.1.2 Applications to GraphTheory I. Self-Study Questions. Exercises. 1.2 Systems. Rank: The Maximum Number of Linearly Independent Equations. True-False Questions. Exercises. 1.2.1 Computer Projects. Exercises. 1.2.2 Applications to Circuit Theory. Self-StudyQuestions. Exercises. 1.3 Gaussian Elimination. Spanning in Polynomial Spaces. Computational Issues: Pivoting. True-False Questions. Exercises. Computational Issues: Flops. 1.3.1 Computer Projects. Exercises. 1.3.2 Applications to Traffic Flow. Self-Study Questions. Exercises. 1.4 Column Space and Nullspace. Subspaces. Subspaces of Functions. True-False Questions. Exercises. 1.4.1 Computer Projects. Exercises. 1.4.2 Applications to Predator-Prey Problems. Self-Study Questions. Exercises. Chapter Summary. 2. Linear Independence and Dimension. 2.1 The Test for Linear Independence. Bases for the Column Space. Testing Functions for Independence. True-False Questions. Exercises. 2.1.1 Computer Projects. 2.2 Dimension. True-False Questions. Exercises. 2.2.1 Computer Projects. Exercises. 2.2.2 Applications to Calculus. Self-Study Questions. Exercises. 2.2.3 Applications to Differential Equations. Self-Study Questions. Exercises.2.2.4 Applications to the Harmonic Oscillator. Self-Study Questions. Exercises. 2.2.5 Computer Projects. Exercises. 2.3 Row Space and the Rank-Nullity Theorem. Bases for the Row Space. Rank-Nullity Theorem. Computational Issues: Computing Rank. True-False Questions. Exercises. 2.3.1 Computer Projects. Chapter Summary. 3. Linear Transformations. 3.1 The Linearity Properties. True-False Questions. Exercises. 3.1.1 Computer Projects. 3.1.2 Applications to Control Theory. Self-Study Questions. Exercises. 3.2 Matrix Multiplication (Composition). Partitioned Matrices. Computational Issues: Parallel Computing. True-False Questions. Exercises. 3.2.1 Computer Projects. 3.2.2 Applications to Graph Theory II. Self-Study Questions. Exercises. 3.3 Inverses. Computational Issues: Reduction vs. Inverses. True-False Questions. Exercises. Ill Conditioned Systems. 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. True-False Questions. Exercises. 3.5.1 Computer Projects. Chapter Summary. 4. Determinants. 4.1 Definition of the Determinant. 4.1.1 The Rest of theProofs. True-False Questions. Exercises. 4.1.2 Computer Projects. 4.2 Reduction and Determinants. Uniqueness of the Determinant. True-False Questions. Exercises. 4.2.1 Application to Volume. Self-Study Questions. Exercises. 4.3 A Formula for Inverses. Cramers Rule. True-False Questions. Exercises 273. Chapter Summary. 5. Eigenvectors and Eigenvalues. 5.1 Eigenvectors. True-False Questions. Exercises. 5.1.1 Computer Projects. 5.1.2 Application to Markov Processes.Exercises. 5.2 Diagonalization. Powers of Matrices. True-False Questions. Exercises. 5.2.1 Computer Projects. 5.2.2 Application to Systems of Differential Equations. Self-Study Questions. Exercises. 5.3 Complex Eigenvectors. Complex Vector Spaces. Exercises. 5.3.1 Computer Projects. Exercises. Chapter Summary.6. Orthogonality. 6.1 The Scalar Product in Rn. Orthogonal/Orthonormal Bases and Coordinates. True-False Questions. Exercises. 6.1.1 Application to Statistics. Self-Study Questions. Exercises. 6.2 Projections: The Gram-Schmidt Process. The QR Decomposition 334. Uniqueness of the QR-factoriaition. True-False Questions. Exercises. 6.2.1 Computer Projects. Exercises. 6.3 Fourier Series: Scalar Product Spaces. Exercises. 6.3.1 Computer Projects. Exercises. 6.4 Orthogonal Matrices. Householder Matrices. True-False Questions. Exercises. 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. True-False Questions. Exercises. 6.6.1 Computer Projects. Exercises. 6.7 The Singular Value Decomposition (SVD). Application of the SVD to Least-Squares Problems. True-False Questions. Exercises. Computing the SVD Using Householder Matrices. Diagonalizing Symmetric Matrices Using Householder Matrices. 6.8 Hermitian Symmetric and Unitary Matrices. True-False Questions. Exercises. Chapter Summary. 7. Generalized Eigenvectors. 7.1Generalized Eigenvectors. Exercises. 7.2 Chain Bases. Jordan Form. True-FalseQuestions. Exercises. The Cayley-Hamilton Theorem. Chapter Summary. 8. Numerical Techniques. 8.1 Condition Number. Norms. Condition Number. Least Squares. Exercises. 8.2 Computing Eigenvalues. Iteration. The QR Method. Exercises. Chapter Summary. Answers and Hints. Index.

• ISBN: 978-0-470-17884-3
• Editorial: John Wiley & Sons
• Encuadernacion: Cartoné
• Páginas: 512
• Fecha Publicación: 04/07/2008
• Nº Volúmenes: 1
• Idioma: Inglés