Find bases for the subspaces $U_1, U_2, U_1 \cap U_2, U_1 + U_2$ Let $$U_1 = \left\{ \left( 
\begin{array}{cc} 
x_1\\ 
x_2\\
x_3\\
x_4\\ 
\end{array} 
\right) \in \mathbb{R} :-x_1-x_2+x_3=0 \right\}$$
$$U_2 = span \left( \left( \begin{array}{cc}
-1  \\
1 \\
1 \\
2 \\ \end{array} \right), \left( \begin{array}{cc}
2 \\
-1 \\
-1 \\
2 \\ \end{array} \right) \right) $$
, find the bases for  $U_1, U_2, U_1 \cap U_2, U_1 + U_2$ and give the dimension of each subspace.
Now $U_1$ can be written as $$\left\{ \left( 
\begin{array}{cc} 
x_1\\ 
x_2\\
x_1 + x_2\\
x_4\\ 
\end{array} 
\right)  \right\}$$ and so a basis would be $$B= \left( \left( \begin{array}{cc}
1  \\
0 \\
1 \\
0 \\ \end{array} \right), \left( \begin{array}{cc}
0 \\
1 \\
1 \\
0 \\ \end{array} \right), \left( \begin{array}{cc}
0 \\
0 \\
0 \\
1 \\ \end{array} \right) \right)$$
the vectors in the span of $U_2$ form a basis since they are linearly independent.
Now to find the basis of $U_1 + U_2$ I could take the span of all the basis vectors in $U_1$ and $U_2$, but since there are only four linearly independent vectors in that span, only four of those vectors form the basis of $U_1 + U_2$.
I cannot find a basis for the intersection of $U_1$ and $U_2$. I can only determine the dimension of that subspace by using the dimension formula for subspaces. So $U_1 \cap U_2$ must have dimension 1, but I cannot find a spanning vector. Can anybody help please?

I tend to use a bare-hands approach like this:
Suppose $x \in U_1 \cap U_2$. Since $x \in U_2$, this implies that 
$$x= s \left( 
\begin{array}{cc} 
-1\\ 
1\\
1\\
2\\ 
\end{array} 
\right)
+t \left( 
\begin{array}{cc} 
2\\ 
-1\\
-1\\
2\\ 
\end{array} 
\right) = 
\left( 
\begin{array}{cc} 
-s + 2t\\ 
s-t\\
s-t\\
2s+2t\\ 
\end{array} 
\right)
$$
Since we require this to belong to $U_1$, we need $x_3 = x_1 + x_2$, so that $s=2t$.
So we can just take (for example) $t=1$ and $s=2$. This gives a vector in the intersection:
$$x = \left( 
\begin{array}{cc} 
0\\ 
1\\
1\\
6\\ 
\end{array} 
\right)$$
Since this is non-zero, it is a spanning set (in fact a basis) for the intersection.
