# vector space

let X is a vector space in feild real number R.
if M, N, K is sub-vector space X and X = M \oplus N = M \oplus K. may we infer that N = K ? if this is wrong in some case, please give me an example shows that.
thanks! i expect your ideas.

### Re: vector space

Suppose (by absurd) that $N \neq K$, then there exists $u \in N$ ($u \neq 0$, $0$ the zero element of the vector space and of course the zero vector of its subspaces) such that $u \notin K$. Then, since $u \in X = M \oplus K = M \oplus N$ the following conditions should hold simoultaneously

$u \notin M$
$u \in N$
$u \notin K$

and the contradiction follows.