Fundamentals of Linear Algebra

Fundamentals of Linear Algebra is like no other book on the subject. By following a natural and unified approach to the subject it has, in less than 250 pages, achieved a more complete coverage of the subject than books with more than twice as many pages. For example, the textbooks in use in the United States prove the existence of a basis only for finite dimensional vector spaces. This book proves it for any given vector space. With his experience in algebraic geometry and commutative algebra, the author defines the dimension of a vector space as its Krull dimension. By doing so, most of the facts about bases when the dimension is finite, are trivial consequences of this definition. To name one, the replacement theorem is no longer needed. It becomes obvious that any two bases of a finite dimensional vector space contain the same number of vectors. Moreover, this definition of the dimension works equally well when the geometric objects are nonlinear. Features: Presents theories and applications in an attempt to raise expectations and outcomes The subject of linear algebra is presented over arbitrary fields Includes many non-trivial examples which address real-world problems

Table of Contents:
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