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Fundamentals of Linear Algebra: Textbooks in Mathematics

Autor J.S. Chahal
en Limba Engleză Paperback – 21 ian 2023
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
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Specificații

ISBN-13: 9781032475813
ISBN-10: 1032475811
Pagini: 240
Ilustrații: 17
Dimensiuni: 156 x 234 mm
Greutate: 0.52 kg
Ediția:1
Editura: CRC Press
Colecția CRC Press
Seria Textbooks in Mathematics

Locul publicării:Boca Raton, United States

Notă biografică

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.

Cuprins

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


 



Descriere

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.