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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!

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2 Answers 2

up vote 4 down vote accepted

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.

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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.. –  Galois Group Aug 6 '12 at 7:39
    
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 6 '12 at 8:05
    
@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 ____" –  Galois Group Aug 12 '12 at 18:33

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.

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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..:) –  Ryan Aug 6 '12 at 7:09
    
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 6 '12 at 7:28
    
That's why I was careful to state "linear space" in my comment above, as per Joachim's answer. –  Ryan Aug 6 '12 at 12:13

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