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.
