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!
Is a 2-dimensional subspace always called a plane no matter what the dimensions of the space is?
2 Answers
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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0Except 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.. – 2012-08-06
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0Well 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. – 2012-08-06
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0@FortuonPaendrag, do you have the same problem with the linear span of a monomial? Polynomial vector spaces of bounded degree are finite-dimensional. – 2012-08-12
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0@alancalvitti : I am unsure, but I would just refer to it as the "subspace spanned by ____" – 2012-08-12
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0Can we call a subspace of Codim=2 a hyperline? – 2018-03-27
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0@john I never heard anyone use that terminology. But in math, you can define your own objects... ;) – 2018-06-19
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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0So 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..:) – 2012-08-06
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0A 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. – 2012-08-06
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0That's why I was careful to state "linear space" in my comment above, as per Joachim's answer. – 2012-08-06