Direct sums

[gravenewworld]

I have to prove or give a counter example to the statement if U1, U2, W are subspaces of V such that V=U1 direct sum W and V=U2 direct sum W, then U2=U1.

This is what I did: Let v be an element of V. Then v=v1+v2 for v1 an element of U1 and v2 and element of W and v=v3+v2 for v3 an element of U2. So v-v2=v1 and v-v2=v3. Therefore v1=v3. Hence U1=U2 since every vector in each subspace is the same.

I just feel like I am missing something to make my small proof 100% airtight. Should I mention somewhere that v is represented in a unique way since V=U1 direct sum W and V=U2 direct sum W?

[Galileo]

This is what I did: Let v be an element of V. Then v=v1+v2 for v1 an element of U1 and v2 and element of W and v=v3+v2 for v3 an element of U2. So v-v2=v1 and v-v2=v3. Therefore v1=v3.
Be careful, how do you know whether your v2 in the expression v=v1+v2 is the same v2 as the one in v=v3+v2?

[gravenewworld]

I guess I could explicity write that v2 is the same vector in both situations.

[gravenewworld]

Actually, doesn't v2 have to be the same for both situations since v is the same? Since V is a direct sum of both subspaces then v has a unique representation so v2, has to be the same right?

[Galileo]

Yes, that's true. But I would write that out explicitly, it doesn't appear trivial to me. I'd use a basis of V to write v, then take those vectors that are in U1 to form a basis for U1, the rest will form a basis for W. Then the result follows from the uniqueness of the basis expansion.