This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LU-factorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.
Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, and more.
This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on understanding the structure of linear operators on vector spaces.
The best way to learn is to do, and the purpose of this book is to get the reader to DO linear algebra.
This text guides the student to try out different programs by providing specific commands.
One of the best available works on matrix theory in the context of modern algebra, this text bridges the gap between ordinary undergraduate studies and completely abstract mathematics. 1952 edition.
Readers of this book will not only come away with the knowledge that the results of linear algebra are true, but also with a deep understanding of why they are true.
(b) Find the Jordan form for D : V → V and a basis that puts [D]xx into this form. Finally, we pose a problem that would be hard to resolve without appealing to the JCF. It has infinitely many solutions, but they break into a finite ...
Just as there is a standard basis for Po, a standard basis also exists for the space M(m, n) of m × n matrices. ... e-| | *-| | Since B is clearly linearly independent, it is a basis (the “standard basis”) for M(2, 2), which we order as ...
(60349-0) Lie Algebras, Nathan Jacobson. (63832-4) Greek Mathematical Thought and the Origin of Algebra, Jacob Klein. (27289-3) Theory and Application of 1nfinite Series, Konrad Knopp. (66165-2) Applied Analysis, Cornelius Lanczos.
A brief introduction to Mathematica is provided in Appendix A.2. As readers work through this book, it is important to understand the basic ideas, definitions, and computational skills.