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.
Work partially supported by DFG Grant Me872/7-3
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Errett Bishop and Douglas S. Bridges. Constructive Analysis, Springer, Berlin, 1985. 451
Vasco Brattka and Klaus Weihrauch. Computability on subsets of Euclidean space I: Closed and compact subsets. Theoretical Computer Science, 219:65–93, 1999. 451, 452, 455, 456, 457
Xiaolin Ge and Anil Nerode. On extreme points of convex compact Turing located sets. InAnil Nerode and Yu. V. Matiyasevich, editors, Logical Foundations of Computer Science, vol. 813 of LNCS, 114–128, Berlin, 1994. Springer. 457
Andrzej Grzegorczyk. On the definitions of computable real continuous functions. Fundamenta Mathematicae, 44:61–71, 1957. 450
Ker-I Ko. Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston, 1991. 450
Daniel Lacombe. Les ensembles récursivement ouverts ou fermés, et leurs applications à l’Analyse récursive. Comp. Rend. Acad. des Sci. Paris, 246:28–31, 1958. 450
Andrè Lieutier. Toward a data type for solid modeling based on domain theory. In K.-I Ko, A. Nerode, M. B. Pour-El, K. Weihrauch, and J. Wiedermann, eds, Computability and Complexity in Analysis, vol. 235 of Informatik Berichte, pages 51–60. FernUniversität Hagen, August 1998. 451, 457
Marian B. Pour-El and J. Ian Richards. Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin, 1989. 450
Alan M. Turing. On computable numbers, with an application to the “Entschei-dungsproblem”. Proceedings of the London Mathematical Society, 42(2):230–265, 1936. 450
Klaus Weihrauch. Computable Analysis. Springer, Berlin, 2000. 450, 451, 452, 453, 456
Ning Zhong. Recursively enumerable subsets of Rq in two computing models: Blum-Shub-Smale machine and Turing machine. Theoretical Computer Science, 197:79–94, 1998. 457
Qing Zhou. Computable real-valued functions on recursive open and closed subsets of Euclidean space. Mathematical Logic Quarterly, 42:379–409, 1996. 457
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ziegler, M., Brattka, V. (2000). Computing the Dimension of Linear Subspaces. In: Hlaváč, V., Jeffery, K.G., Wiedermann, J. (eds) SOFSEM 2000: Theory and Practice of Informatics. SOFSEM 2000. Lecture Notes in Computer Science, vol 1963. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44411-4_34
Download citation
DOI: https://doi.org/10.1007/3-540-44411-4_34
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41348-6
Online ISBN: 978-3-540-44411-4
eBook Packages: Springer Book Archive