An Introduction to Linear Algebra and Tensors
Courier Corporation, 1972 - Mathematics - 167 pages
The present book, a valuable addition to the English-language literature on linear algebra and tensors, constitutes a lucid, eminently readable and completely elementary introduction to this field of mathematics. A special merit of the book is its free use of tensor notation, in particular the Einstein summation convention. The treatment is virtually self-contained. In fact, the mathematical background assumed on the part of the reader hardly exceeds a smattering of calculus and a casual acquaintance with determinants.
The authors begin with linear spaces, starting with basic concepts and ending with topics in analytic geometry. They then treat multilinear forms and tensors (linear and bilinear forms, general definition of a tensor, algebraic operations on tensors, symmetric and antisymmetric tensors, etc.), and linear transformation (again basic concepts, the matrix and multiplication of linear transformations, inverse transformations and matrices, groups and subgroups, etc.). The last chapter deals with further topics in the field: eigenvectors and eigenvalues, matrix polynomials and the Hamilton-Cayley theorem, reduction of a quadratic form to canonical form, representation of a nonsingular transformation, and more. Each individual section -- there are 25 in all -- contains a problem set, making a total of over 250 problems, all carefully selected and matched. Hints and answers to most of the problems can be found at the end of the book.
Dr. Silverman has revised the text and numerous pedagogical and mathematical improvements, and restyled the language so that it is even more readable. With its clear exposition, many relevant and interesting problems, ample illustrations, index and bibliography, this book will be useful in the classroom or for self-study as an excellent introduction to the important subjects of linear algebra and tensors.
Unabridged and unaltered republication of revised English edition originally titled "Introductory Linear Algebra, " 1972.
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MULTILINEAR FORMS AND TENSORS
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angle antisymmetric arbitrary vectors basis vectors Bibliography bilinear form CALCULUS called carrying the vector characteristic equation characteristic surface coefficients collinear component form concisely considered in Example coordinate system defined degree not exceeding denoted DIFFERENTIAL EQUATIONs dimension eigenvalues eigenvectors elements equals expansion form of degree formula function geometrically given hence homothetic transformation identity transformation INTRODUCTION invariant inverse linear space linear subspaces linearly dependent linearly independent vectors mathematical mation multilinear form multiplication nonnegative nonzero null space operation origin orthogonal matrix orthogonal transformation orthonormal basis parallelepiped perpendicular plane L2 polynomials of degree Prob problems Proof Prove quadratic form radius vector rank real number Remark respect rotation scalar product scalar triple product second-order tensor solution space C[a space L2 summation Suppose symmetric linear transformation symmetric transformation tensor determined tensor of order theorem theory trilinear vector arguments vector lying vector product vector x e write zero