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Textbook
Linear Algebra: Ideas and Applications, 3rd EditionISBN: 978-0-470-17884-3
Hardcover
504 pages
July 2008, ©2008
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Features of the Text.
1. Systems of Linear Equations.
1.1 The Vector Space of m x n Matrices.
The Space R<sup>n</sup>.
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 R<sup>n</sup>.
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.
