A SYSTEM OF ELEMENTARY ALGEBRA
1 LINEAR SYSTEMS OF EQUATIONS 1 11 Some Examples 1 12 Notation and a Review of Numbers 12 13 Gaussian Elimination: Basic Ideas 24 14 Gaussian Elimination: General Procedure 37 15 *Applications and Computational Notes 52 16 *Projects and Reports 61 2 MATRIX ALGEBRA 65 21 Matrix Addition and Scalar Multiplication 65 22 Matrix Multiplication 72 23 Applications of Matrix Arithmetic 83 24 Special Matrices and Transposes 103 25 Matrix Inverses 118 26 Determinants 141 27 *Tensor Products 160 28 *Applications and Computational Notes 166 29 *Projects and Reports 177 3 VECTOR SPACES 181 31 Definitions and Basic Concepts 181 32 Subspaces 198 33 Linear Combinations 206 34 Subspaces Associated with Matrices and Operators 220 35 Bases and Dimension 229 36 Linear Systems Revisited 239 37 *Change of Basis and Linear Operators 248 38 *Introduction to Linear Programming 254 39 *Applications and Computational Notes 273 310 *Projects and Reports 274 4 GEOMETRICAL ASPECTS OF STANDARD SPACES 277 41 Standard Norm and Inner Product 277 42 Applications of Norms and Vector Products 288 43 Orthogonal and Unitary Matrices 302 44 *Applications and Computational Notes 314 45 *Projects and Reports 327 5 THE EIGENVALUE PROBLEM 331 51 Definitions and Basic Properties 331 52 Similarity and Diagonalization 343 53 Applications to Discrete Dynamical Systems 354 54 Orthogonal Diagonalization 366 55 *Schur Form and Applications 372 56 *The Singular Value Decomposition 375 57 *Applications and Computational Notes 379 58 *Project Topics 386 6 GEOMETRICAL ASPECTS OF ABSTRACT SPACES 391 61 Normed Spaces 391 62 Inner Product Spaces 398 63 Orthogonal Vectors and Projection 410 64 Linear Systems Revisited 418 65 *Operator Norms 424 66 *Applications and Computational Notes 431 67 *Projects and Reports 442