Fundamentals of Linear Algebra

Preface Advice to the Reader 1 Preliminaries What is Linear Algebra? Rudimentary Set Theory Cartesian Products Relations Concept of a Function Composite Functions Fields of Scalars Techniques for Proving Theorems 2 Matrix Algebra Matrix Operations Geometric Meaning of a Matrix Equation Systems of Linear Equation Inverse of a Matrix The Equation Ax=b Basic Applications 3 Vector Spaces The Concept of a Vector Space Subspaces The Dimension of a Vector Space Linear Independence Application of Knowing dim (V) Coordinates Rank of a Matrix 4 Linear Maps Linear Maps Properties of Linear Maps Matrix of a Linear Map Matrix Algebra and Algebra of Linear Maps Linear Functionals and Duality Equivalence and Similarity Application to Higher Order Differential Equations 5 Determinants Motivation Properties of Determinants Existence and Uniqueness of Determinant Computational Definition of Determinant Evaluation of Determinants Adjoint and Cramer's Rule 6 Diagonalization Motivation Eigenvalues and Eigenvectors Cayley-Hamilton Theorem 7 Inner Product Spaces Inner Product Fourier Series Orthogonal and Orthonormal Sets Gram-Schmidt Process Orthogonal Projections on Subspaces 8 Linear Algebra over Complex Numbers Algebra of Complex Numbers Diagonalization of Matrices with Complex Eigenvalues Matrices over Complex Numbers 9 Orthonormal Diagonalization Motivational Introduction Matrix Representation of a Quadratic Form Spectral Decompostion Constrained Optimization-Extrema of Spectrum Singular Value Decomposition (SVD) 10 Selected Applications of Linear Algebra System of First Order Linear Differential Equations Multivariable Calculus Special Theory of Relativity Cryptography Solving Famous Problems from Greek Geometry Answers to Selected Numberical Problems Bibliography Index