Matrices and Vector SPates

William C. Brown, Michigan State University

• I. MATRICES AND LINEAR SYSTEMS OF EQUATIONS 1. Matrices 2. Matrix Arithmetic 3. The Multiplication of Partitioned Matrices 4. Gaussian Elimination 5. Elementary Matrices and Inverses 6 LU Factorizations II. VECTOR SPACES 1. Definitions and Examples 2. Finite Dimensional Vector Spaces, Bases, and Dimension 3. The Rank of a Matrix 4. Linear Transformations 5. Matrix Representations of Linear Transformations III. DETERMINANTS 1. Permutations and the Definition of the Determinant 2. Basic Theorems on Determinants 3. The Laplace Expansion 4. Eigenvalues and the Spectrum of a Matrix IV. INNER PRODUCT SPACES 1. Real Inner Product Spaces 2. Complex Inner Product Spaces 3. Least Squares Problems 4. Normal Matrices5. Real Quadratic Forms