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

ISBN-10: 0486635457

ISBN-13: 9780486635453

Trans. by means of Richard A. Silverman

The authors commence with linear areas, beginning with uncomplicated innovations and finishing with themes in analytic geometry. They then deal with multilinear types and tensors (linear and bilinear types, normal definition of a tensor, algebraic operations on tensors, symmetric and antisymmetric tensors, etc.), and linear transformation (again simple ideas, the matrix and multiplication of linear modifications, inverse ameliorations and matrices, teams and subgroups, etc.). The final bankruptcy bargains with additional issues 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. every one person part — there are 25 in all — features a challenge set, creating a overall of over 250 difficulties, all conscientiously chosen and coupled. tricks and solutions to many of the difficulties are available on the finish of the book.

Dr. Silverman has revised the textual content and various pedagogical and mathematical advancements, and restyled the language in order that it truly is much more readable. With its transparent exposition, many correct and engaging difficulties, plentiful illustrations, index and bibliography, this ebook may be worthwhile within the school room or for self-study as an exceptional advent to the $64000 matters of linear algebra and tensors.

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**Additional resources for An Introduction to Linear Algebra and Tensors**

**Sample text**

3. The definition of a multilinear form q> = (p(x, y, z , . . , w , . . , the value of tp depends only on the values of its vector arguments. For example, a trilinear form (p = (p{x, y, z) depends only on the values of the vectors x, y, z and not on the components of x, y, z relative to any underlying basis e2, e3. In the language o f p. 25, we can say that multilinear forms have been defined in an invariant fashion. Since the components of a vector change in transforming to a new basis, the same must be true of the coefficients of a multilinear form (if the form itself is to remain invariant).

Proof For simplicity, consider the special case p = 3, q = 2, where the set of numbers (1) is of the form aijklm. Suppose the quantity = S ijk a ijk lm h m is a tensor whenever tlrr is a tensor. Let tlm = utvm (the product of two arbitrary vectors ut and vm). Then S ijk = a iJ k lm U i°m '> and contracting this expression with arbitrary vectors x n y p z k, we get W A = *iJkimXiyJZkUiVm. 54 CHAP. 2 MULTILINEAR FORMS AND TENSORS Since siJk is a tensor, the expression on the left is a scalar. It follows that the expression on the right, which depends linearly on the compo nents of the vectors x, y, z, u, v, is a multilinear form of degree 5.

If two conditions of the form 1 2 ) 0 >(x, y, z t + z2, . . , w) = p(x, y, z„ . . , w) +

(x, y, Az,. . , w) = Ap(x, y, z, . . , w) hold for each of the arguments x, y, z , . . , w. The number of arguments p is called the degree of the multilinear form, and

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

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