Hi guys,
I have a problem from my homework and am not sure where to start. (I hate proof questions). Could some1 just point me in the direction, I'm sure I can do it myself once I get some pointers, thanks!
You better get over hating proofs, because that's pretty much what math is.
Clearly each of the . If it were also in , we would get , which would imply that the set is not linearly independent, contradicting the fact that it is a basis for something.
Thus, for each , . Since subspaces are closed under taking linear combinations of their elements, this implies .
For the second part,
.
But since we already know that the intersection of these two subspaces is trivial, this means that the sum is actually a direct sum: .