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Say $U\leq V$ for a vector space $V$, and $v \space \in \space V$. We define an affine subspace to be a coset of a vector subspace, namely $$v+U=\{v+u | u \space\epsilon\space U\} \subset V$$

However it can be proven that this is only a subspace when $v \space\in\space U$.

So my question is, in the case when $v \not\in U$, why is the coset of the vector subspace still called an affine subspace when it is not actually a vector space?

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The phrase "affine subspace" has to be read as a single term. It refers, as you said, to a coset of a subspace of a vector space. As is common in mathematics, this does not mean that an "affine subspace" is a "subspace" that happens to be "affine" - an "affine subspace" is usually not a subspace at all.

But there is a relationship between affine subspaces and (genuine) subspaces, which is the motivation for the terminology. Every subspace is an affine subspace, and every affine subspace can be translated to become a subspace. So, geometrically, subspaces and affine subspaces have a similar structure.

There are many other examples in mathematics where this kind of thing happens - where a modifier to a word creates a compound term which refers to something that the original word did not refer to. Another example is a "skew field", which is not usually a field.

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  • $\begingroup$ Could you add a little more on the relationship between affine subspaces and genuine subspaces? $\endgroup$ – Thomas Smith Dec 26 '17 at 12:56

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