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Any general rules as to when an affine space is a vector subspace and when it's not?

Because Wikipedia example:

https://en.wikipedia.org/wiki/Affine_space#/media/File:Affine_space_R3.png

says that in that case $P_2$ is not a subspace.

However the article gives in many parts that an affine space is a subspace. However, does this not even imply that it could be a vector/linear subspace in some cases?


The article also write:

Any vector space may be considered as an affine space

So this means that affine spaces can be vector spaces without the null vector property?

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An affine space is a linear subspace if and only if the affine space contains the null vector.

The nomenclature makes sense if you think about an affine function. If it goes through 0, it is a linear function.

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  • $\begingroup$ But is there some name for a subspace that has the other properties, but just not the null property? Because that's useful as well. $\endgroup$ – mavavilj Jul 5 '18 at 20:02
  • $\begingroup$ Which other properties? A linear subspace $V$ has the property that if $v \in V$, then $cv \in V \; \forall c \in \mathbb{R}$, so this property overlaps with the null property and gives you vectors arbitrarily close to 0. Maybe you are looking for a convex pointed cone without 0, where the aforementioned property only has to hold for $c>0$? $\endgroup$ – LinAlg Jul 5 '18 at 20:06
  • $\begingroup$ @ Linearity without $0$ vector. That for $a,b,$ in the affine space, $c(a+b)$ $(c>0)$ is also there. Or at least $a+b$. $\endgroup$ – mavavilj Jul 5 '18 at 20:10
  • $\begingroup$ That would be a convex pointed cone without 0 $\endgroup$ – LinAlg Jul 5 '18 at 20:12

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