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Linear algebra : ideas and applications / Richard Penney.

By: Material type: TextTextPublication details: Hoboken, N.J. : John Wiley, c2008.Edition: 3rd edDescription: xvi, 480 p. ill. ; 25 cmISBN:
  • 9780470178843 (hbk.)
  • 0470178841 (hbk.)
Subject(s): DDC classification:
  • 512/.5 22 P4137
Contents:
Preface. Features of the Text. 1. Systems of Linear Equations. 1.1 The 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 Graph Theory 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-Study Questions. 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 the Proofs. 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. Cramer’s 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. 7.1 Generalized Eigenvectors. Exercises. 7.2 Chain Bases. Jordan Form. True-False Questions. 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.
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Includes index.

Preface.
Features of the Text.
1. Systems of Linear Equations.
1.1 The 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 Graph Theory 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-Study Questions.
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 the Proofs.
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.
Cramer’s 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.
7.1 Generalized Eigenvectors.
Exercises.
7.2 Chain Bases.
Jordan Form.
True-False Questions.
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.

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