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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Direct sums

**Physics Forums | Science Articles, Homework Help, Discussion**