Subspace intersecting many other subspaces V is a vector space of dimension 7. There are 5 subspaces of dimension four. I want to find a two dimensional subspace such that it intersects at least once with all the 5 subspaces.
Edit: All the 5 given subspaces are chosen randomly (with a very high probability, the intersection is a line).
If i take any two of the 5 subspaces and find the intersection it results in a line. Similarly, we can take another two planes and find another line. From these two lines we can form a 2 dimensional subspace which intersect 4 of the 5 subspaces. But can some one tell me how we can find a two dimensional subspace which intersect all the 5 subspace.
It would be very useful if you can tell what kind of concepts in mathematics can i look for to solve problems like this?
Thanks in advance.
Edit: the second paragraph is one way in which i tried the problem. But taking the intersection of the subspace puts more constraint on the problem and the solution becomes infeasible.

By "intersects" do you mean that the intersection is nonzero? Then in general there won't be a solution: the two-dimensional subspace spanned by the intersection of subspaces 1 and 2 and the intersection of subspaces 3 and 4 will intersect subspace 5 only in the zero vector.
For example, if $e_1, \ldots, e_7$ form a basis of $V$ take $U_1$ spanned by $e_1, e_2, e_3, e_4$, $U_2$ spanned by $e_1, e_5, e_6, e_7$, $U_3 = U_1$, $U_4$ spanned by $e_2, e_5, e_6, e_7$, and $U_5$ spanned by $e_3, e_4, e_5, e_6$.    
Edit:  Oops, this example is bogus, please ignore.
