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 .

so:

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!