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Warn moderators| Author : | Topic: multilinear GR | Bottom |
| very very small tic Posts : 70  |
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| ferme Posts : 85 |
** Linear algebra, sometimes disguised as matrix theory, considers sets and functions which preserve linear structure. In practice this includes a very wide portion of mathematics! Thus linear algebra includes axiomatic treatments, computational matters, algebraic structures, and even parts of geometry; moreover, it provides tools used for analyzing differential equations, statistical processes, and even physical phenomena. ** http://www.math.niu.edu/~rusin/known-math/index/15-XX.html |
| saucer admin Posts : 673 A Good Tautology is Hard to Find!  |
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| zee Posts : 115 |
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| zee Posts : 115 |
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| zee Posts : 115 |
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| zee Posts : 115 |
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| zee Posts : 115 |
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| saucer admin Posts : 673 A Good Tautology is Hard to Find! |
- Historical background of the approach to multilinear algebra The subject itself has various roots going back to the mathematics of the nineteenth century, in what was then called tensor analysis, or the "tensor calculus of tensor fields". It developed out of the use of tensors in differential geometry, general relativity, and many branches of applied mathematics. Around the middle of the 20th century the study of tensors was reformulated more abstractly. The Bourbaki group's treatise Multilinear Algebra was especially influential — in fact the term multilinear algebra was probably coined there. One reason at the time was a new area of application, homological algebra. The development of algebraic topology during the 1940s gave additional incentive for the development of a purely algebraic treatment of the tensor product. The computation of the homology groups of the product of two spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion (see Künneth theorem). The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined. The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to De Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product. The resulting rather severe write-up of the topic (by Bourbaki) entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups). They instead applied a novel approach using category theory, with the Lie group approach viewed as a separate matter. Since this leads to a much cleaner treatment, there was probably no going back in purely mathematical terms. (Strictly, the universal property approach was invoked; this is somewhat more general than category theory, and the relationship between the two as alternate ways was also being clarified, at the same time.) Indeed what was done is almost precisely to explain that tensor spaces are the constructions required to reduce multilinear problems to linear problems. This purely algebraic attack conveys no geometric intuition. Its benefit is that by re-expressing problems in terms of multilinear algebra, there is a clear and well-defined 'best solution': the constraints the solution exerts are exactly those you need in practice. In general there is no need to invoke any ad hoc construction, geometric idea, or recourse to co-ordinate systems. In the category-theoretic jargon, everything is entirely natural. - |
| 3141582n Posts : 6  |
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