A subvector space, or subspace, of \(\RNrSpc{n}\) is a subset \(V\) of \(\RNrSpc{n}\) with the following properties

1. The \(\Vect{0}\)-vector belongs to \(V\).
2. \(V\) is closed under vector addition; that is, if \(\Vect{x}\) and \(\Vect{y}\) are in \(V\), then \(\Vect{x}+\Vect{y}\) is also in \(V\).
3. \(V\) is closed under scalar multiplication; that is, if \(\Vect{x}\) is in \(V\) and \(t\) is a number in \(\RNr\), then \(t \Vect{x}\) is in \(V\).

If the context is clear we may say ‘subspace’ instead of ‘subvector space’. Thus, if \(V,W\) are subspaces of \(\RNrSpc{n}\), we say that \(V\) is a subspace of \(W\) if \(V\) is contained in \(W\).