Computing the Dimension of Linear Subspaces

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.