Vector space and maximal proper subspace
Suppose that V is a vector space over F and is any nonzero vector.
Show that there is a subspace W of V maximal subject to the condition
that ; show that any such W is a maximal proper subspace of V .
(Do NOT assume that V is ﬁnite-dimensional.)
For the first part I thought that I could say :
1. Prove that we can exchange one of the vector in basis of V by .
2. Prove that the basis (B) is still one.
4. is a basis and thus a maximal independant subset that doesn't contain .
5. W is a subspace of V.
Is it correct?
For the second part I don't have a good idea of what to do.
Any hint would be great. Thanks!