Preface

1. Systems of Equations

* Solutions of systems of equations

2. Augmented Matrices and Elementary Row Operations

* Augmented matrix *
Elementary row operation *
Echelon form *
Reduced echelon form *
Gauss-Jordan elimination *
Transpose *
Rank

3. The Algebra of Matrices

* Matrix addition *
Scalar multiplication *
Matrix multiplication

4. Inverses of Matrices

* Matrix inversion

5. Determinants, Adjoints, and Cramer's Rule

* Determinant *
Adjoint *
Cramer's rule

6. Application: Matrix Algebra and Modular Arithmetic

* Modular arithmetic *
Matrix operations *
Hill codes

7. Vector Products, Lines, and Planes

* Dot product *
Cross product *
Projection *
Unit vector *
Vectors in R?n *
Orthogonal vectors

8. Vector Spaces and Subspaces

* Vector space *
Subspace *
Spaces of functions and matrices *
Linear combination *
Spanning Set *
Null Space *
Rank of a matrix

9. Independence, Basis and Dimension

* Linearly independent set *
Basis *
Dimension *
Coordinate vector

10. Row Space, Column Space, and Null Space

* Row space *
Column space *
Null space *
Rank *
Nullity

11. Inner Product Spaces

* General inner products

12. Orthonormal Bases and the Gram-Schmidt Process

* Orthonormal basis *
Gram-Schmidt process

13. Change of Basis and Orthogonal Matrices

* Transition (or change of basis) matrix *
Orthogonal matrices

14. Eigenvalues and Eigenvectors

* Characteristic polynomial *
Eigenvalue *
Eigenvector *
Eigenspace

15. Diagonalization and Orthogonal Diagonalization

*
Similarity *
Diagonalization *
Symmetric matrix *
Orthogonal diagonalization

16. Matrices and Linear Transformations from R?m to R?n

* Linear transformation *
Matrix of a linear transformation *
Kernel of a linear transformation *
Image of a linear transformation *
Inverse of a linear transformation *
Composition of linear transformations

17. Matrices of General Linear Transformations;
Similarity

* Matrix of a linear transformation *
Similar matrices

18. Applications and Numerical Methods

* Systems of differential equations *
Gauss-Seidel method *
Generalized inverse and curve fitting *
Rotation of axes *
LU and QR factorizations

Appendix A. Maple V Mini-Reference

Appendix B. User-Defined Functions