In linear algebra, vectors $ v_1, v_2... v_n $ form a basis for the subspace V when
- The vectors span V. (in other words, the span of the vectors is V)
- The vectors are linearly independent.
If V is a subspace of $ \mathbb R^m $ it follows that n must be less than or equal to m.
Note that there can be more than one set of vectors that form a basis for the same space. In fact, there are an infinite number of bases (plural of basis) for a subspace provided the subspace is not just $ \vec 0 $. However, all bases for a given subspace have the same number of vectors. This number of vectors is called the dimension of the subspace.
