Let $\displaystyle V $ be the set of vectors in $\displaystyle \ \Re^4 $ of the form $\displaystyle (x, x+y, x-y, y) $ where $\displaystyle x, y \ \epsilon \ \Re $. Find a subspace $\displaystyle W \ \epsilon \ \Re^4 $ such that $\displaystyle \ \Re^4 = V \oplus W $.

Attempt at a solution:

Now it is easy to prove that $\displaystyle V $ is of dimension 2 with a basis as $\displaystyle [(1,1,-1,0), \ (0,1,1,1)] $. All we need is a 2 dimensional subspace $\displaystyle W $ such that the basis of both $\displaystyle V \mbox{and} \ W $ have no element in common.

By trial, $\displaystyle W $ with a basis as $\displaystyle [(1,1,1,1), \ (1,0,1,0)] $ satisfies the given condition. My question is, is there a systematic method to solve such questions? What do I do if the basis of the involved vector spaces are so high that trial and error don't work?

Thanks!