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
