Finding a Basis for R4 with Given Data

[Dell]

i am given a subspace of R⁴ W={(a b c d)} and know a+c=0 c-2d=0, and am asked to find a basis for W,
i wrote (-2 0 2 1)(0 1 0 0),
now i am asked to find the missing vectors so that the new basis will be a basis for R⁴. to find this i need vectors that are independent and are the basis for R⁴-W? how can i do this??
just a guess would be (-2 0 2 1)(0 1 0 0)(0 0 1 0)(0 0 0 1) but that's just because i know that those four would cover the whole space and are independent. what is the mathematical way of solving this?

 

[Mark44]

Gram-Schmidt process --http://en.wikipedia.org/wiki/Gram_schmidt

 

[Dell]

another question, not related,
is it possible for 2 separate basises of a subspace to be linearly independent of one another, or do they always need to be independent

 

[Mark44]

I don't think it's meaningful to talk about one basis being independent of another. For linear independence/dependence, we're always talking about a collection of vectors.

Suppose v1, v2, ... , vn are a basis for a subspace W and u1, u2, ..., un are another basis for W. The set of vectors {v1, v2, ..., vn, u1} has to be a linearly dependent set, meaning that u1 has to be a linear combination of v1, v2, ... , vn.

A basis for a subspace W is the largest set of vectors that a) is linearly independent, and b) spans W. If you add any vector to this basis, the added vector must be a linear combination of the original basis vectors.