Let be subspaces.
i need to prove that if and then .
here's the thing.
i proved it, but without considering .
(i proved that if then )
and that's obviously can't be, i mean there must be something i'm doing wrong, i just can't figure out what.
here's what i did:
to prove that U and W are the direct sum of V, i just need to show that for every : there's only one linear combination of and .
Let be the basis for U.
Let be the basis for W.
now, since , i can conclude that the number of vectors in V's basis must be .
so, the basis for V can now be:
now, let's presume v can be presented in two different ways, and show that it's actually the same presentation, so:
let's presume there are scalars , not all 0, and , not all 0, such that:
if we'll subtruct one from another we'll get:
and since are linear independent (basis for V), then , and so on...
so that's all.
my question now is: where exactly does fit in? why do i even need it here?
it's quite a task to use the math terminology when it's not your native language, so i hope i used it right and everything is clear enough...
thanks in advanced!