Let U and V be subspaces of a vector space W. If W=U [tex]\oplus[/tex] V, show U [tex]\bigcap[/tex] V={0}. I'm a bit lost on this one... as I thought this was essentially the definition of direct sum. I'm unsure where to start. Any help would be great!
I wasn't sure if that was necessary info or not... looks like I was wrong :) If U and V are subspaces of vector space W, and each w in W can be written uniquely as a sum u+v where u is in U and v is in V then W is a direct sum of U and V.
Well, surely you can do something on it -- even if it's just rewriting the problem in a less opaque form. e.g. do you know anything about proving two subspaces equal? (Or two sets?)
I think you should add the definition as follows: If U and V are subspaces of a Vector space W and each [tex]w \in W[/tex] can be written as the unique sum as u+v where [tex]u \in U[/tex] and [tex]v \in V[/tex] then W is the direct sum of U and V and can be written [tex]W = U \oplus V[/tex]
Suppose there were a non-zero vector, w, in both U and V and let u be any vector in U. Now, write u as two different sums of vectors in U and V.