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
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
Biography
Dr. J.S. Chahal is a professor of mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University and after spending a couple of years at the University of Wisconsin as a post doc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes and has published a number of papers about number theory. For hobbies, he likes to travel and hike, the reason he accepted the position at Brigham Young University.
