An Introduction to Linear Algebra and Tensors

M. A. Akivis, V. V. Goldberg, Richard A. Silverman

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486148786
• Topic: Mathematics
• Subtopic: Linear Algebra
• Index: Mathematics

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 ploynomials 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.

1

LINEAR SPACES

1. Basic Concepts

In studying analytic geometry, the reader has undoubtedly already encountered the concept of a free vector, i.e., a directed line segment which can be shifted in space parallel to its original direction. Such vectors are usually denoted by boldface Roman letters like a, b, . . . , x, y, . . . It can be assumed for simplicity that the vectors all have the same initial point, which we denote by the letter 0 and call the origin of coordinates.

Two operations on vectors are defined in analytic geometry:

1. Any two vectors x and y can be added (in that order), giving the sum x + y;
2. Any vector x and (real) number a can be multiplied, giving the product λ•x or simply λx.

The set of all spatial vectors is closed with respect to these two operations, in the sense that the sum of two vectors and the product of a vector with a number are themselves both vectors.

The operations of addition of vectors x, y, z, . . . and multiplication of vectors by real numbers λ, μ, . . . have the following properties:

1. x + y = y + x;
2. (x + y) + z = x + (y + z);
3. There exists a zero vector 0 such that x + 0 = x;
4. Every vector x has a negative (vector) y = − x such that x + y = 0;
5. 1•x = x;
6. λ(μx) = (λμ)x;
7. (λ + μ)x = λx + μx;
8. λ(x + y) = λx + λy.

However, operations of addition and multiplication by numbers can be defined for sets of elements other than the set of spatial vectors, such that the sets are closed with respect to the operations and the operations satisfy the properties 1)–8) just listed. Any such set of elements is called a linear space (or vector space), conventionally denoted by the letter L. The elements of a vector space L are often called vectors, by analogy with the case of ordinary vectors.

Example 1. The set of all vectors lying on a given straight line l forms a linear space, since the sum of two such vectors and the product of such a vector wi...