Linear Algebra with Applications

1. Linear Equations

1.1 Introduction to Linear Systems

1.2 Matrices, Vectors, and Gauss-Jordan Elimination

1.3 On the Solutions of Linear Systems; Matrix Algebra



2. Linear Transformations

2.1 Introduction to Linear Transformations and Their Inverses

2.2 Linear Transformations in Geometry

2.3 Matrix Products

2.4 The Inverse of a Linear Transformation



3. Subspaces of R" and Their Dimensions

3.1 Image and Kernel of a Linear Transformation

3.2 Subspace of R"; Bases and Linear Independence

3.3 The Dimension of a Subspace of R"

3.4 Coordinates



4. Linear Spaces

4.1 Introduction to Linear Spaces

4.2 Linear Transformations and Isomorphisms

4.3 The Matrix of a Linear Transformation



5. Orthogonality and Least Squares

5.1 Orthogonal Projections and Orthonormal Bases

5.2 Gram-Schmidt Process and QR Factorization

5.3 Orthogonal Transformations and Orthogonal Matrices

5.4 Least Squares and Data Fitting

5.5 Inner Product Spaces



6. Determinants

6.1 Introduction to Determinants

6.2 Properties of the Determinant

6.3 Geometrical Interpretations of the Determinant; Cramer's Rule



7. Eigenvalues and Eigenvectors

7.1 Dynamical Systems and Eigenvectors: An Introductory Example

7.2 Finding the Eigenvalues of a Matrix

7.3 Finding the Eigenvectors of a Matrix

7.4 Diagonalization

7.5 Complex Eigenvalues

7.6 Stability



8. Symmetric Matrices and Quadratic Forms

8.1 Symmetric Matrices

8.2 Quadratic Forms

8.3 Singular Values



9. Linear Differential Equations

9.1 An Introduction to Continuous Dynamical Systems

9.2 The Complex Case: Euler's Formula

9.3 Linear Differential Operators and Linear Differential Equations



Appendix A. Vectors

Answers to Odd-numbered Exercises

Subject Index

Name Index