Definition 1. 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 $X = v + W$.
Proposition 1. Every subspace of a vector space is an affine subspace.
Proposition 2. Let $V$ be a vector space. For any affine subspace $X = v + W$ ($v \in V$, $W \subseteq V$):
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For any $x \in V$,
\[x \in X \enspace\lrimp\enspace x - v \in W.\] -
For any $x \in X$,
\[X = x + W.\]
Proposition 3. The underlying subspace of every affine subspace is unique. That means, if $V$ is a vector space, $v$, $v’ \in V$ and $W$, $W’ \subseteq V$ are subspaces such that
\[v + W = v' + W',\]then $W = W’$.