vector space

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# vector space

Submitted by nona_nonomo on Fri, 11/12/2010 - 18:36

Forums:

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.

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## Versions

(v1) by nona_nonomo 2010-11-12

## 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.