Groups, Matrices, And Vector Spaces: A Group Theoretic Approach To Linear Algebra (universitext)
by James B. Carrell /
2017 / English / PDF
4.3 MB Download
This unique text provides a geometric approach to group
theory and linear algebra, bringing to light the interesting ways
in which these subjects interact. Requiring few prerequisites
beyond understanding the notion of a proof, the text aims to give
students a strong foundation in both geometry and
algebra. Starting with preliminaries (relations, elementary
combinatorics, and induction), the book then proceeds to the core
topics: the elements of the theory of groups and fields
(Lagrange's Theorem, cosets, the complex numbers and the prime
fields), matrix theory and matrix groups, determinants, vector
spaces, linear mappings, eigentheory and diagonalization, Jordan
decomposition and normal form, normal matrices, and quadratic
forms. The final two chapters consist of a more intensive look at
group theory, emphasizing orbit stabilizer methods, and an
introduction to linear algebraic groups, which enriches the
notion of a matrix group.
This unique text provides a geometric approach to group
theory and linear algebra, bringing to light the interesting ways
in which these subjects interact. Requiring few prerequisites
beyond understanding the notion of a proof, the text aims to give
students a strong foundation in both geometry and
algebra. Starting with preliminaries (relations, elementary
combinatorics, and induction), the book then proceeds to the core
topics: the elements of the theory of groups and fields
(Lagrange's Theorem, cosets, the complex numbers and the prime
fields), matrix theory and matrix groups, determinants, vector
spaces, linear mappings, eigentheory and diagonalization, Jordan
decomposition and normal form, normal matrices, and quadratic
forms. The final two chapters consist of a more intensive look at
group theory, emphasizing orbit stabilizer methods, and an
introduction to linear algebraic groups, which enriches the
notion of a matrix group.
Applications involving symm
Applications involving symmetry groups, determinants, linear coding theory and
cryptography are interwoven throughout. Each section ends with
ample practice problems assisting the reader to better understand
the material. Some of the applications are illustrated
in the chapter appendices. The author's unique melding of
topics evolved from a two semester course that he taught at the
University of British Columbia consisting of an undergraduate
honors course on abstract linear algebra and a similar course on
the theory of groups. The combined content from both makes this
rare text ideal for a year-long course, covering more material than
most linear algebra texts. It is also optimal for independent study
and as a supplementary text for various professional applications.
Advanced undergraduate or graduate students in mathematics,
physics, computer science and engineering will find this book both
useful and enjoyable.
etry groups, determinants, linear coding theory and
cryptography are interwoven throughout. Each section ends with
ample practice problems assisting the reader to better understand
the material. Some of the applications are illustrated
in the chapter appendices. The author's unique melding of
topics evolved from a two semester course that he taught at the
University of British Columbia consisting of an undergraduate
honors course on abstract linear algebra and a similar course on
the theory of groups. The combined content from both makes this
rare text ideal for a year-long course, covering more material than
most linear algebra texts. It is also optimal for independent study
and as a supplementary text for various professional applications.
Advanced undergraduate or graduate students in mathematics,
physics, computer science and engineering will find this book both
useful and enjoyable.