# Subspaces

Part of a vector space spanned by a given spanning set.

The subspace of a vector space \(\mathcal{V}\) is a subset of \(\mathcal{V}\) is also a vector space.

E.g. span of a set of vectors of \(\mathcal{V}\).

## Definition

If \(S\) is a nonempty subset of a vector space \(V\), and \(S\) satisfies the conditions:

- \(\alpha |x\rangle\in S\) whenever \(|x\rangle\in S\) for any scalar \(\alpha\)
- \(|x\rangle + |y\rangle\in S\) whenever \(|x\rangle\in S\) and \(|y\rangle\in S\)

then \(S\) is said to be a subspace of \(V\).