Linear algebra : ideas and applications / Richard C . Penney, Purdue University.

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