Theorem. Let S be a subset of a vector space V and let a be an element in S which is equal to a linear combination of other elements of S. Let S' be the set obtained by removing a from S. Then span(S)=span(S').

Proof. We use the theorem about spans. By this theorem, we need to show that every element in S' is a linear combination of elements in S and every element in S is a linear combination of elements from S'.

The first statement is clear because every element s in S' belongs to S, so it is a "one-term" linear combination of elements of S: s=s.

In order to prove the second statement take an arbitrary element s from S.

If s is not equal to a then it belongs to S' and so s is a "one term" linear combination of elements of S'.

If s=a then by the condition of our theorem, s is a linear combination of elements from S which are distinct from a, that is s is a linear combination of elements of S'. Thus every element of S is a linear combination of elements in S'.