Let $V$ be a vector space over a field $F$ and let $U$ be a subspace of $V$
a) Prove that the quotient $V/U$ has a natural structure of a vector space over $F$
b) Prove that $dim U + dim V/U = dim V$
My attempt: Every vector space has a basis so let {$V_1,V_2,..., V_n$} be a basis for $U$. Then this basis is a linearly independent set of vectors of $V$ so $V$ has a basis that contains {$V_1,V_2, ..., V_n$}. Now when you take the quotient $V/U$ you are left with the basis elements {$V_{n+1}, V_{n+2}, ...$}. I don't know where to go from here. A friend told me that I just have to show that $V/U$ inherits a natural basis, but I don't understand what he meant. I think what I wrote above is sufficient for showing (b)