# Linnear Algebra Isomorphisms : prove that f + g is an isomorphism?

1. Feb 13, 2010

### zeion

1. The problem statement, all variables and given/known data

Suppose f and g are isomorphisms from U to V. Prove of disprove each of the following statements:
a) The mapping f + g is an isomorphism from U to V.

2. Relevant equations

3. The attempt at a solution

I have no idea where to start.. do I need to show that f and g are 1-1 and onto?
Or do I go from something like f(u) + g(u) = (f+g)(u)?
Isn't that defined by a property of composition of function..?

2. Feb 14, 2010

### CompuChip

Note that isomorphism is a stronger property than bijection (1-1 and onto).
If h is an isomorphism, it means that h(u + u') = h(u) + h(u') for all u and u' in U.
So you should how this for h = (f + g).

3. Feb 14, 2010

### Hurkyl

Staff Emeritus
There's a classic math problem: find two irrational numbers x and y such that x+y is rational. This often gives people a lot of trouble, because they spend all their effort trying to guess irrational x and y then checking if x+y is rational.