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Is a 2-dimensional subspace in a 7-dimensional space still called a plane? I know that a 6-dimensional space in 7-dimensional space is called a hyperplane because the difference in the number of dimensions of the space and subspace is 1. The answer should be easily googlable, but for some reason it's eluding me. Thanks!

ryang
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2 Answers2

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i guess this is all convention, but i feel safe to say:

Name of linear spaces (i.e. not curved):

  • Dim=1: line

  • Dim=2: plane

  • Codim=1: hyperplane

When the space is not linear:

  • Dim=1: curve

  • Dim=2: surface

  • Codim=1: hypersurface.

Codimenion is just a name for that difference in dimension you mentioned. So a hyperplane in a 2 dimensional space is in fact a line, even weirder a hyperplane in a 1 dimensional space is a point... When the hypersurface is given by a polynomial of degree $d$, it is common to refer to it as quadric ($d=2$), cubic ($d=3$), etc.

Edit: I claim no knowledge of terminology when the spaces are infinite dimensional.

Joachim
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  • Except for example, in the space of all continuous real valued functions on $\mathbb R$, it would be weird to call the subspace spanned by $e^x$ a line.. – Arkady Aug 06 '12 at 07:39
  • Well for infinite dimensional spaces i don't know the convention. Thanks, i added an edit. But for me, it would make sense to call the $\mathbb{R}$-span of $e^x$ a line; you need one parameter to specify a point on it. – Joachim Aug 06 '12 at 08:05
  • @FortuonPaendrag, do you have the same problem with the linear span of a monomial? Polynomial vector spaces of bounded degree are finite-dimensional. – alancalvitti Aug 12 '12 at 15:26
  • @alancalvitti : I am unsure, but I would just refer to it as the "subspace spanned by ____" – Arkady Aug 12 '12 at 18:33
  • Can we call a subspace of Codim=2 a hyperline? – john Mar 27 '18 at 09:10
  • @john I never heard anyone use that terminology. But in math, you can define your own objects... ;) – Joachim Jun 19 '18 at 13:22
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From my understanding it is. It is analogous to points and lines, which also just convey a concept invariant of the dimensionality of the space they are embedded in. I think hyperplane is a more confusing term because it is not a plan and presumably is called hyperplane because it separates a n-dimensional space into to parts and thus the actual subspace it describes depends on the dimensionality of the space.

But of course the two-dimensional sub-space has to be flat in order for it to be a plane.

simmmons
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  • So a 2-dimensional subspace (plane) is always flat in a linear space of any dimension then. I.e. is a plane still flat even in a 8-dimensional space. Sounds obvious now, but I wasn't so sure before..:) – ryang Aug 06 '12 at 07:09
  • A 2-dimensional subspace does not need to be flat but if it is it is a plane otherwise a surface (well a plane is also a surface but not the other way round). The dimensionality of the encompassing space does not matter but please see Joachim's answer, he did a better job clarifying the terminology. – simmmons Aug 06 '12 at 07:28
  • That's why I was careful to state "linear space" in my comment above, as per Joachim's answer. – ryang Aug 06 '12 at 12:13