Understanding affine subsets I'm learning linear algebra and am having some questions about affine subsets.
An affine subset is defined (in Linear Algebra Done Right 3th edition) as a subset of vector space $V$, that can be expressed as $v+U$, where $v\in V$, $U$ is a subspace of $V$.
Now, how exactly is it defined? Is a subspace $U$ of $V$ given first, then I define all sets parallel to to $U$? or $U$ is just arbitrary?
If the latter is true, can I say all subspaces are affine subsets? Because, 
I can always write $U-z=\{u-z:u\in U \}$, where $z \in U$, and clearly $U-z$ is a subspace. Then I can write $U = (U-z)+z$. 
Thanks!
 A: Let $W$ be a subset of $V$.  Then $W$ is an "affine subset" of $V$ if and only if there exists a vector $v \in V$ and a subspace $U$ of $V$ such that $W = v + U$.
It follows from this definition that any subspace of $V$ is an affine subset of $V$.  Proof: If $W$ is a subspace of $V$, then $W$ can be written as $W = v + U$, where $v = 0$ and $U = W$.
A: Let $U$ be a subspace of $V$. According to the definition, all cosets of the form $u+U$ are affine. Conversely, let $A$ be the affine set. Then there exists $u\in V$ s.t. $U:=-u+A$ is a subspace of $V$. So, having the definition of an affine set, we can construct the appropriate parallel subspace.
Of course, all subspaces are affne, it is enough to take $u=0$.
My argument written above is expressed in another words in the last sentences of your post.
Similarly to cosets of a group we could introduce the equivalence relation in $V$. Given a subspace $U$. We say that $$w\sim v\iff v-w\in U.$$ The cosets wrt. this relation are exactly all affine subsets parallel to $U$.
