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Computing the Dimension of Linear Subspaces
Martin Ziegler and Vasco Brattka, in:
SOFSEM 2000: Theory and Practice of Informatics,
Lecture Notes in Computer Science 1963, 450-458,
Springer, Berlin, 2000.
Abstract
Since its very beginning, linear algebra is a highly
algorithmic subject. Let us just mention the famous Gauß
Algorithm which was invented before the theory of algorithms
has been developed. The purpose of this paper is to link
linear algebra explicitly to computable analysis, that is the
theory of computable real number functions.
Especially, we will investigate in which sense the dimension
of a given linear subspace can be computed.
The answer highly depends on how the linear subspace is given:
if it is given by a finite number of vectors whose linear
span represents the space, then the dimension does not depend
continuously on these vectors and consequently it cannot
be computed. If the linear subspace is represented via
its distance function, which is a standard way to represent
closed subspaces in computable analysis, then the dimension
does computably depend on the distance function.
Electronical Versions
© 2000 Springer-Verlag Berlin