An Introduction to Linear Algebra and Tensors by M. A. Akivis, V. V. Goldberg, Richard A. Silverman

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

Trans. by way of Richard A. Silverman

The current booklet, a worthy addition to the English-language literature on linear algebra and tensors, constitutes a lucid, eminently readable and entirely hassle-free creation to this box of arithmetic. a unique benefit of the booklet is its loose use of tensor notation, particularly the Einstein summation conference. The therapy is nearly self-contained. in truth, the mathematical history assumed at the a part of the reader rarely exceeds a smattering of calculus and an informal acquaintance with determinants.
The authors commence with linear areas, beginning with simple thoughts and finishing with issues in analytic geometry. They then deal with multilinear kinds and tensors (linear and bilinear types, common definition of a tensor, algebraic operations on tensors, symmetric and antisymmetric tensors, etc.), and linear transformation (again uncomplicated thoughts, the matrix and multiplication of linear variations, inverse modifications and matrices, teams and subgroups, etc.). The final bankruptcy offers with additional themes within the box: eigenvectors and eigenvalues, matrix ploynomials and the Hamilton-Cayley theorem, aid of a quadratic shape to canonical shape, illustration of a nonsingular transformation, and extra. each one person part — there are 25 in all — encompasses a challenge set, creating a overall of over 250 difficulties, all rigorously chosen and coupled. tricks and solutions to lots of the difficulties are available on the finish of the book.
Dr. Silverman has revised the textual content and diverse pedagogical and mathematical advancements, and restyled the language in order that it truly is much more readable. With its transparent exposition, many proper and engaging difficulties, considerable illustrations, index and bibliography, this e-book should be necessary within the school room or for self-study as an exceptional advent to the real topics of linear algebra and tensors.

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The components o f the tensor aijk determined by q> do not change under arbitrary permutations of indices. 2. A bilinear form q> = p(x, y) is said to be antisymmetric if «K*. y) - -

5. Contraction of products of tensors. , the tensors aijk and blm (of orders 3 and 2, respectively), suppose we form the product aijkblm (a tensor of order 5), and then contract the resulting tensor with respect to the indices k and /, say. This gives a tensor aijkbkm = ai]\b\m + aij2^2m of order 3, and the corresponding operation is again called contraction, more exactly, contraction of the tensors aijk and blm with respect to the indices k and /. Thus the operation of contracting two tensors consists of 53 ALGEBRAIC OPERATIONS ON TENSORS SEC.

Then the parametric equations of / can be written in the form Xk = x l + X(IJkajl)a ^ (k = 1, 2, 3). 6. General equation of a second-degree curve in the plane. The general equation of a second-degree curve relative to some rectangular coordinate system in the plane is given by A x 2 + 2Bxy + Cy2 + 2 D x + 2Ey + F = 0. (9) Let the coordinates x and y be denoted by x l and x 2. Moreover, let au denote the coefficient of the product let at denote the coefficient of x i9 and let a denote the constant term.

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