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Edition: 93

Copyright: 1993

Publisher: Wellesley-Cambridge Press

Published: 1993

International: No

Copyright: 1993

Publisher: Wellesley-Cambridge Press

Published: 1993

International: No

**1 Introduction**

1.1 Vectors and Matrices

1.2 Lengths and Dot Products

**2 Solving Linear Equations**

2.1 Linear Equations

2.2 The Idea of Elimination

2.3 Elimination Using Matrices

2.4 Rules for Matrix Operations

2.5 Inverse Matrices

2.6 Elimination = Factorization: A = LU

2.7 Transposes and Permutations

**3 Vector Spaces and Subspaces**

3.1 Spaces of Vectors

3.2 The Nullspace of A: Solving Ax = 0

3.3 The Rank and the Row Reduced Form

3.4 The Complete Solution to Ax=b

3.5 Independence, Basis, and Dimension

3.6 Dimensions of the Four Subspaces

**4 Orthogonality**

4.1 Orthogonality of the Four Subspaces

4.2 Projections

4.3 Least Squares Approximations

4.4 Orthogonal Bases and Gram-Schmidt

**5 Determinants**

5.1 The Properties of Determinants

5.2 Permutations and Cofactors

5.3 Cramer's Rule, Inverses, and Volumes

**6 Eigenvalues and Eigenvectors**

6.1 Introduction to Eigenvalues

6.2 Diagonalizing a Matrix

6.3 Applications to Differential Equations

6.4 Symmetric Matrices

6.5 Positive Definite Matrices

6.6 Similar Matrices

6.7 The Singular Value Decomposition

**7 Linear Transformations**

7.1 The Idea of a Linear Transformation

7.2 The Matrix of a Linear Transformation

7.3 Change of Basis

7.4 Diagonalization and the Pseudoinverse

**8 Applications**

8.1 Graphs and Networks

8.2 Markov Matrices and Economic Models

8.3 Linear Programming

8.4 Fourier Series: Linear Algebra for Functions

8.5 Computer Graphics

**9 Numerical Linear Algebra**

9.1 Gaussian Elimination in Practice

9.2 Norms and Condition Numbers

9.3 Iterative Methods for Linear Algebra

**10 Complex Vectors and Complex Matrices**

10.1 Complex Numbers

10.2 Hermitian and Unitary Matrices

10.3 The Fast Fourier Transform

Solutions to Selected Exercises

Index

Copyright: 1993

Publisher: Wellesley-Cambridge Press

Published: 1993

International: No

Table of Contents

**1 Introduction**

1.1 Vectors and Matrices

1.2 Lengths and Dot Products

**2 Solving Linear Equations**

2.1 Linear Equations

2.2 The Idea of Elimination

2.3 Elimination Using Matrices

2.4 Rules for Matrix Operations

2.5 Inverse Matrices

2.6 Elimination = Factorization: A = LU

2.7 Transposes and Permutations

**3 Vector Spaces and Subspaces**

3.1 Spaces of Vectors

3.2 The Nullspace of A: Solving Ax = 0

3.3 The Rank and the Row Reduced Form

3.4 The Complete Solution to Ax=b

3.5 Independence, Basis, and Dimension

3.6 Dimensions of the Four Subspaces

**4 Orthogonality**

4.1 Orthogonality of the Four Subspaces

4.2 Projections

4.3 Least Squares Approximations

4.4 Orthogonal Bases and Gram-Schmidt

**5 Determinants**

5.1 The Properties of Determinants

5.2 Permutations and Cofactors

5.3 Cramer's Rule, Inverses, and Volumes

**6 Eigenvalues and Eigenvectors**

6.1 Introduction to Eigenvalues

6.2 Diagonalizing a Matrix

6.3 Applications to Differential Equations

6.4 Symmetric Matrices

6.5 Positive Definite Matrices

6.6 Similar Matrices

6.7 The Singular Value Decomposition

**7 Linear Transformations**

7.1 The Idea of a Linear Transformation

7.2 The Matrix of a Linear Transformation

7.3 Change of Basis

7.4 Diagonalization and the Pseudoinverse

**8 Applications**

8.1 Graphs and Networks

8.2 Markov Matrices and Economic Models

8.3 Linear Programming

8.4 Fourier Series: Linear Algebra for Functions

8.5 Computer Graphics

**9 Numerical Linear Algebra**

9.1 Gaussian Elimination in Practice

9.2 Norms and Condition Numbers

9.3 Iterative Methods for Linear Algebra

**10 Complex Vectors and Complex Matrices**

10.1 Complex Numbers

10.2 Hermitian and Unitary Matrices

10.3 The Fast Fourier Transform

Solutions to Selected Exercises

Index