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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Images of bilinear mappings into $\mathbf R^3$


Authors: S. J. Bernau and Piotr J. Wojciechowski
Journal: Proc. Amer. Math. Soc. 124 (1996), 3605-3612
MSC (1991): Primary 15A69
MathSciNet review: 1340376
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Abstract: It is well-known that the image of a multilinear mapping into a vector space need not be a subspace of its target space. It is, however, far from clear which subsets of the target space may be such images. For vector spaces over the real numbers we give a complete classification of the images of bilinear mappings into a three-dimensional vector space. In Theorem 2.8 we show that either the image of a bilinear mapping into a three-dimensional space is a subspace, or its complement is either the interior of a double elliptic cone, or a plane from which two lines intersecting at the origin have been removed. We also show (Theorem 2.2) that the image of any multilinear mapping into a two-dimensional space is necessarily a subspace. Our methods are elementary and free of tensor considerations.


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Additional Information

S. J. Bernau
Affiliation: Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, Texas 79968-0514
Address at time of publication: College of Science, California State Polytechnic University, 3801 W. Temple Avenue, Pomona, California 91768-4031
Email: sjbernau@csupomona.edu

Piotr J. Wojciechowski
Affiliation: Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, Texas 79968-0514
Email: piotr@math.utep.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03432-6
PII: S 0002-9939(96)03432-6
Received by editor(s): January 24, 1995
Received by editor(s) in revised form: May 22, 1995
Communicated by: Lance W. Small
Article copyright: © Copyright 1996 American Mathematical Society