What does it mean to say a subspace has a "non-zero constant"?

Do you mean to say the L and P have a non-zero vector in common? That is, that there exist at least one non-zero vector in both P and Q.

If so, construct a basis for each of P and Q. If L and P have no non-zero vectors in common, then the union of the two bases is still independent and so span a subspace of dimension 6, which is impossible.