By M. A. Akivis, V. V. Goldberg, Richard A. Silverman
Trans. through Richard A. Silverman
The authors commence with linear areas, beginning with easy options and finishing with themes in analytic geometry. They then deal with multilinear kinds and tensors (linear and bilinear varieties, basic definition of a tensor, algebraic operations on tensors, symmetric and antisymmetric tensors, etc.), and linear transformation (again easy techniques, the matrix and multiplication of linear changes, inverse differences 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, relief of a quadratic shape to canonical shape, illustration of a nonsingular transformation, and extra. every one person part — there are 25 in all — includes 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 fascinating difficulties, plentiful illustrations, index and bibliography, this booklet could be worthwhile within the school room or for self-study as a very good creation to the $64000 matters of linear algebra and tensors.
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Additional resources for An Introduction to Linear Algebra and Tensors
That its value depends only on the vectors x, y and not on the choice of basis. Under a change of basis q>goes over into
. By formula (2) above and formula (9), p. 26, tp = atr x t y r = yi'iyj'jaijy ,kx kyriy l = yriytkyn yrf i iJx ky lMoreover, 7nVn< = <5,t , yfJyf , = Sj„ by the properties of an orthogonal matrix,! and hence
6 BASIS TRANSFORMATIONS. TENSOR CALCULUS 25 with determinant 1. In the second case, the transformation formulas are of the form e r = e t cos 0 + e 2 sin 0 , er = e x sin 6 — e 2 cos 0 , so that r becomes cos 0 sin 0 sin 0 —cos 6 t with determinant —1 . 2. Consider- any spatial vector x. , a force, velocity, acceleration, or electric field intensity. This “real” object does not depend on the coordinate system in which it is considered, and hence any operations or calculations directly involving vectors must always have a physical interpretation.
2 MULTILINEAR FORMS AND TENSORS formula (9), p. 26. , the components of the tensor determined by the form, must also transform in some perfectly well-defined way. This transformation law is given by the following T heorem . m depending on the choice o f basis forms a tensor i f and only i f they transform according to the law ^ i'j'k '- - m y V i ? j ‘j l l k ' k * V m ' n f l i j k - • -m under the transformation from one orthonormal basis e 1 5 e2, e 3 to another orthonormal basis ev, er >er .
An Introduction to Linear Algebra and Tensors by M. A. Akivis, V. V. Goldberg, Richard A. Silverman