Proof regarding Range of linear maps

[ricramos]

Homework Statement

Prove that if T, S : V -> W are linear and W is finite dimensional then
dim (range(S + T)) <= dim range S + dim range T

Homework Equations

The Attempt at a Solution

I know that the dim range S + dim range T can be at most 2 * dim W. However, I'm a little stuck on trying to figure out what dim (range(S+T)) is. How do I determine S + T?

 

[HallsofIvy]

Suppose Range S and Range T have only the 0 vector in common. What is S+ T? If they have more than the 0 vector in common, their intersection must be a subspace.

Or is the problem that you do not know how S+ T is defined? It is the space of all vectors orf the form u+ v where u is in S and v is in T.