Fundamentals of Linear Algebra - J.S. Chahal

Fundamentals of Linear Algebra

(Autor)

Buch | Softcover
240 Seiten
2023
Chapman & Hall/CRC (Verlag)
978-1-032-47581-3 (ISBN)
54,85 inkl. MwSt
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

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.

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

Erscheinungsdatum
Reihe/Serie Textbooks in Mathematics
Zusatzinfo 17 Illustrations, black and white
Sprache englisch
Maße 156 x 234 mm
Gewicht 331 g
Themenwelt Mathematik / Informatik Mathematik Algebra
Mathematik / Informatik Mathematik Angewandte Mathematik
Mathematik / Informatik Mathematik Logik / Mengenlehre
ISBN-10 1-032-47581-1 / 1032475811
ISBN-13 978-1-032-47581-3 / 9781032475813
Zustand Neuware
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