Prove or disprove
are subspaces of .
If
then
Attempt:
False
let's say
and
then may equal 4
and may equal 4
and then we get
The misconception that some students have, and that this problem is attempting to dispel, is that the direct sum is like a (set) union. But in fact, the direct sum of U1 and U2 contains a whole lot of elements that are not in U1 or U2. In case Tonio's excellent example wasn't clear, think about this and read it again.