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

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?

• 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 $c>0$? – LinAlg Jul 5 '18 at 20:06
• @ Linearity without $0$ vector. That for $a,b,$ in the affine space, $c(a+b)$ $(c>0)$ is also there. Or at least $a+b$. – mavavilj Jul 5 '18 at 20:10