###### Definition 3.5.7

A *basis* for a subspace \(S\) of \(\real^p\) is a set of vectors in \(S\) that are linearly independent and span \(S\text{.}\) It can be seen that any two bases have the same number of vectors. Therefore, we say that the *dimension* of the subspace \(S\text{,}\) denoted \(\dim
S\text{,}\) is the number of vectors in any basis.