Let V be the subset of R5 consisting of all quintuples (a1,a2,a3,a4,a5) such that
a1+a2+a3+a4+a5=0
and
a1+2a2+3a3+4a4+5a5=0
Prove that V is a subspace of R5. Solve the above equations for a1 and a2 in terms of a3,a4 and a5. Hence, exhibit a basis for V, explaining carefully why the vectors you give form a basis.