The collection of vectors in a subvector space \(V\) of \(\RNrSpc{n}\) is closed under scalar multiplication. Here are some illustrations of what this means.
The subset below is closed under vector addition and scalar multiplication. So it is a subspace of \(\RNrSpc{n}\).
The two lines below form a set \(S\) which is closed under scalar multiplication: For any vector \(\Vect{x}\) in \(S\), the entire line through the origin in the direction of \(\Vect{x}\) also belongs to \(V\).
The set \(S\) above is, however, not closed under vector addition. Therefore it is not a subvector space of \(\RNrSpc{n}\).
The blue dots below form a set \(S\) which is closed under vector addition but not under scalar multiplication. To see that it is not closed under scalar multiplication, consider any vector in \(S\): there are points on the line through \(\Vect{x}\) which do not belong to \(S\).
Therefore this set \(S\) is not a subvector space.