Talk:Hyperplane

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Subspace?

There is a mistake in the first line of text: a hyperplane is not necessarily a subspace because it does not necessarily contain the zero element. Should one replace "subspace" with "subset"? — Preceding unsigned comment added by 137.215.6.53 (talk) 08:08, 6 October 2014 (UTC)

It's not a mistake, look at the technical description. Subspace, as used here, is more general than what you are thinking it is. For instance, affine subspaces which generally do not contain the zero element are translates of a vector subspace which does contain the zero element. Bill Cherowitzo (talk) 03:32, 7 October 2014 (UTC)
Agreed. It is a linear subspace which contains the zero. We also have affine subspace and topological subspaces where the vector space structure does not apply.Rgdboer (talk) 20:26, 7 October 2014 (UTC)
Upon reconsideration, 137.215.6.53 has a point. Dealing in dimensions means we have a vector space or manifold. But general manifolds do not have hyperplanes, so the article is in a context of vector spaces. That means a subspace is a linear subspace, the kind passing through the origin. It is recommended that subset be substituted for subspace in the description of a hyperplane.Rgdboer (talk) 02:45, 8 October 2014 (UTC)
Dealing with dimension does not restrict one to vector spaces or manifolds. Affine subspaces and projective subspaces are not linear subspaces, but they are subspaces by the definitions within their disciplines. Perhaps some better definitions and citations for this article would make this point a bit clearer. Bill Cherowitzo (talk) 03:17, 8 October 2014 (UTC)

Related words

Are there words of for flat subspaces of dimension less than n-1? For instance would a flat n-2 subspace be a "hyperline"? --Bequw (talk) 19:36, 1 August 2012 (UTC)

None that I know of in common use besides "point", "line" and "plane" (of dimensions 0, 1 and 2). Generally, flats of dimension k are called k-flats. The term "hyperplane" is only used in reference to (n-1)-flats in n dimensional space. Justin W Smith (talk) 19:51, 1 August 2012 (UTC)
k-planes, k<n-1, have been called "thins" (e.g. Olshevsky's Multimensional Glossary). I have no idea how widespread this is. —Tamfang (talk) 23:09, 1 August 2012 (UTC)