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 simple thoughts 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 options, the matrix and multiplication of linear modifications, inverse modifications 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 — includes a challenge set, creating a overall of over 250 difficulties, all rigorously chosen and paired. tricks and solutions to many 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's much more readable. With its transparent exposition, many correct and fascinating difficulties, abundant illustrations, index and bibliography, this booklet can be valuable within the lecture room or for self-study as a great advent to the real topics of linear algebra and tensors.

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**Extra info for An Introduction to Linear Algebra and Tensors**

**Example text**

17) under transformation to a new basis. ) (see p. 19). Since ei' = Vefit> = 7rfij> e*' 7k'kek, it follows that e i ' j' k ' — 7 ¡ 'i7 j'j7 k 'k ^ ijk ' Remark. In particular, t v v v — 7vi7vj7vk£nk'> • (1 1 ) where there are only six nonzero terms in the right-hand side, so that, in expanded form, ( 1 1 ) becomes * w v = (7vi7r27r3 + 7vi7vz7yx + 7v*7v\7vi — 7 vi7vi7y3 — 7v*7vi7v\ — 7i'i72'37y2)£i23• The quantity in parentheses is just the determinant of the transformation matrix (6 ), and hence Ci'2'3' = ^ 1 23 det r .

1 where x = x' + p, y = y' + p, Z = z' + p. It follows that (3) is invariant under shifts. 2. Equation of a plane. Let II be a plane in space, and let n be a vector normal to II. e,.. (4) Let x 0 be the radius vector of a fixed point P0 e II, with components and let x be the radius vector of an arbitrary vector P e II, with components xf. )er (5) Since the vectors P0P and n are perpendicular, we have P0f - n = 0 , or (x — x0)«n = 0, (6) a result known as the vector form of the equation of II. ) = 0 , or aix i + 6 = (6 ') 0 after denoting —ape?

Then / is determined by the system of two equations a\l)x t + 6 (1>= 0 , a}2>xt + b<2>= 0 , (8 ) where a{(1) and aj2) are components of normal vectors n, and n 2 to the planes H j and I I 2, respectively. To go from (8 ) to (7'), we must find a point PQ on 34 LINEAR SPACES CHAP. 1 / and a vector a parallel to /. Being perpendicular to the vectors n x and n2, the vector a can be chosen as the vector product of n 1 and n2: a = n, x n 2 = euka^af> ek. To find P 0 we need only fix one of the coordinates x t and then solve the system (8 ) for the other two coordinates (one coordinate must be fixed if the system is to have a solution).

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

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