### Combinatorics. An Introduction

- Language: English
- 328 Pages
- Published: March 2013

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- ID: 2182297
- August 2008
- 504 Pages
- John Wiley and Sons Ltd

This expanded new edition presents a thorough and up-to-date introduction to the study of linear algebra

Linear Algebra, Third Edition provides a unified introduction to linear algebra while reinforcing and emphasizing a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, the book successfully helps readers to understand not only how to implement a technique, but why its use is important.

The book outlines an analytical, algebraic, and geometric discussion of the provided definitions, theorems, and proofs. For each concept, an abstract foundation is presented together with its computational output, and this parallel structure clearly and immediately illustrates the relationship between the theory and its appropriate applications. The Third Edition also features:

- A new chapter on generalized eigenvectors and chain bases with coverage of the Jordan form and the Cayley-Hamilton theorem

- A new chapter on numerical techniques, including a discussion of the condition number

- A new section on Hermitian symmetric and unitary matrices

- An exploration of computational
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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.

"Linear Algebra (third edition) is an excellent undergraduate-level textbook for courses in linear algebra. It is also valuable self-study guide for professionals and researches who would like a basic introduction to linear algebra with applications in science, engineering, and computer science." (Mathematical Review, Issue 2009e)

"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 applications." (Electric Review, November 2008)

"This book should make a good text for introductory courses." (Computing Reviews, September 30, 2008)

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