Finding the basis of an intersection of subspaces 
We have subspaces in $\mathbb R^4: $ 
$w_1= \operatorname{sp} \left\{ 
\begin{pmatrix} 1\\  1 \\ 0 \\1 \end{pmatrix} ,
\begin{pmatrix} 1\\  0 \\ 2 \\0 \end{pmatrix},
\begin{pmatrix} 0\\  2 \\ 1 \\1 \end{pmatrix} \right\}$, 
  $w_2= \operatorname{sp} \left\{
\begin{pmatrix} 1\\  1 \\ 1 \\1 \end{pmatrix} ,
\begin{pmatrix} 3\\  2 \\ 3 \\2 \end{pmatrix},
\begin{pmatrix} 2\\  -1 \\ 2 \\0 \end{pmatrix} \right\}$
Find the basis of $w_1+w_2$ and the basis of $w_1\cap w_2$.

So in order to find the basis for $w_1+w_2$, I need to make a $4\times 6$ matrix of all the six vectors, bring it to RREF and see which vector is LD and the basis would be the LI vectors. 
But the intersection of these 2 spans seems empty, or are they the LD vectors that I should've found before ?
In general, how is the intersection of subspaces defined ?

Hint: the intersection of these two spans is NOT empty. What you need to do is find a new spanning set for $w_2$ that contains some of the vectors from the spanning set for $w_1$. The common vectors will span the intersection. Now that you have a basis for $w_1\cap w_2$, you can extend it to a basis of $w_1+w_2$ by adding on the vectors that are in $w_1$ or $w_2$ but not in $w_1\cap w_2$.
