Find a basis for $U\cap V$ 
Let $$a = (0,2,3,-1)^T \quad b=(0,2,7,-2)^T \quad c = (0,-2,1,0)^T \quad u = (1,2,0,1)^T\quad v = (2,2,1,2)^T$$
Let $U= \langle a,b,c \rangle, V = \langle u,v\rangle$
Then a) find a basis for $U$ and $V$, b) find a basis for $U\cap V$.

I have solved a) and found a basis $\{a,b\}$ for $U$ and $\{u,v\}$ for V. But How can I combine this information to find a basis for $U\cap V$?
I have tried:
Let $x \in U\cap V$, then $x = \lambda_1 a+\lambda_2 b = \mu_1u+\mu_2 v$.
Solving $\lambda_1 a + \lambda_2 b - \mu_1 u - \mu_2 v = 0$ results in
$$\lambda_1 = -2r\quad \lambda_2 = r\quad \mu_1 = -2r\quad \mu_2 = r\quad (\forall r \in  \mathbb{Q})$$
This means $a,b,u, v$ are lineair dependent. But now what? I know I can remove one basisvector and this would result in set of independent basis vectors, but doesn't that break the spanning property?
I'm somewhat confused, could someone clarify, no solutions please.

You might want to check out the Zassenhaus algorithm. It gives you not only a basis for $U\cap V$ but also for $U+V$.
