About the intersection of two vector bundles I´m looking for information about the intersection of two vector bundles (principally trivial bundles, but no necessarily). I´m trying to make a picture (literally) of reflexive finite generated modules.

Another related topic is sub-budles of a vector bundles.

All suggestions are wellcome!

Edit: I try to be more specific.
Suppose two sub-bundles of a bundle over a topological space X. We can do the intersection of their vector fibers in each point of base space and collect this intersection vector fibers along the base space. Then we can give it a topology, restriction of the bigger vector bundle. 
What´s about of this "submodule of sections"? 
Is it another vector bundle? 
Has another interesting structure?
 A: You need to refine the question to get better answers, but here are some thoughts:
1) Over a normal variety, you can think of line bundles as divisors, and "intersect" them.
2) A vector bundle can be represented by a reflexive sheaf, but being reflexive is a lot weaker. 
A: More specific than Ilya's answer.  To see that the intersection of two subbundles need not be a bundle, take, on your space $M$, the rank two trivial bundle $M\times \mathbb{R}^2$.  Then take two line bundles $L=(m,x,0)|m\in M, x\in\mathbb{R}$ and let $K$ be a nontrivial line bundle contained in $M\times\mathbb{R}^2$ such that the fiber over some point $m_0$ is the $y=0$ line.  Then away from that point, the intersection will be just the origin, however, at $m_0$, the intersection is a rank 1 vector space.  Thus, the dimension can jump, and so you don't get an actual vector bundle, merely a family of vector subspaces of $\mathbb{R}^2$ parameterized by $M$.  And for this, all we need is some manifold which has a nontrivial subbundle of $\mathbb{R}^2$.
A: The intersection of two subbundles K, L of a bundle M doesn't have to be bundle itself: its dimension can jump up on some algebraic submanifold. The reason why the dimension can go only up is quite simple to visualize by thinking about the intersection of two manifolds EK, EL (total spaces of bundles K, L) in a manifold EM.  
