linear subspace

A linear subspace is a vector space that is a subset of some larger vector space.

Formally,
If $V$ is a vector space over a field $K$,
a subset $W\subseteq V$ is a linear subspace of $V$ if it is a vector space over $K$ for the operations of $V$.

Equivalently, a linear subspace $W$ of $V$ is a nonempty subset of $V$ such that whenever $w_1,w_2$ are elements of $W$ and $\alpha ,\beta$ are elements of $K$, it follows that $\alpha w_1+\beta w_2$ is in $W$.