Let $V$ be a vector space and $X\subseteq V$. $X$ is an affine subspace if there is $v\in V$ and a subspace $W\subseteq V$ such that $X = v + W$.
Every subspace of a vector space is an affine subspace.
Let $V$ be a vector space. For any affine subspace $X = v + W$ ($v\in V$, $W\subseteq V$):
- For any $x\in V$, $x\in X$ if and only if $x - v \in W$.
- $X = x + W$ for all $x\in X$.
The underlying subspace of every affine subspace is unique. That means, if $V$ is a vector space, $v$, $vā\in V$ and $W$, $Wā\subseteq V$ are subspaces such that
\[v + W = v' + W',\]then $W = Wā$.