Let $\displaystyle W,U\subset V$ be subspaces.

i need to prove that if $\displaystyle dim(V)=dim(U)+dim(W)$ and $\displaystyle W\cap U=\left \{ \vec0 \right \}$ then $\displaystyle W\oplus U=V$.

here's the thing.

i proved it, but without considering $\displaystyle W\cap U=\left \{ \vec0 \right \}$.

(i proved that if $\displaystyle dim(V)=dim(U)+dim(W)$ then $\displaystyle W\oplus U=V$)

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 $\displaystyle v\in V$: there's only one linear combination of $\displaystyle u\in U$ and $\displaystyle w\in W$.

so:

Let $\displaystyle \left \{ u_1, u_2... u_n \right \}$ be the basis for U.

Let $\displaystyle \left \{ w_1, w_2... w_m \right \}$ be the basis for W.

now, since $\displaystyle dim(V)=dim(U)+dim(W)$, i can conclude that the number of vectors in V's basis must be $\displaystyle n+m$.

so, the basis for V can now be:

$\displaystyle \left \{ u_1, u_2... u_n,w_1, w_2... w_m \right \}$

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 $\displaystyle a_1,a_2... a_{n+m}\in \mathbb{R}$, not all 0, and $\displaystyle b_1,b_2... b_{n+m}\in \mathbb{R}$, not all 0, such that:

$\displaystyle a_1u_1+a_2u_2+...+a_nu_n+a_{n+1}w_1+a_{n+2}w_2+... +a_mw_m=v$

$\displaystyle b_1u_1+b_2u_2+...+b_nu_n+b_{n+1}w_1+b_{n+2}w_2+... +b_mw_m=v$

if we'll subtruct one from another we'll get:

$\displaystyle (a_1-b_1)u_1+(a_2-b_2)u_2+...+(a_{n}-b_{n})u_n+(a_{n+1}-b_{n+1})w_1+(a_{n+2}-b_{n+2})w_2+...+(a_m-b_m)w_m=0$

and since $\displaystyle \left \{ u_1, u_2... u_n,w_1, w_2... w_m \right \}$ are linear independent (basis for V), then $\displaystyle a_1=b_1$, $\displaystyle a_2=b_2$ and so on...

so that's all.

my question now is: where exactly does $\displaystyle W\cap U=\left \{ \vec0 \right \}$ 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!