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)
You are in a good way for (b)
As you said take $\{v_1,..., v_n\}$ a basis of $U$ and expand this $\{v_1,...,v_n,v_{n+1},...v_{n+k}\}$ in a basis of $V$.
Now show that $\{v_{n+1}+U,..., v_{n+k}+U\}$ is a basis of the quotient $V/U$.