Basis for a subspace I need to calculate the basis for
$$W = \lbrace (a,b,c,d) \: : \:  a+b+c = 0 \rbrace.$$ 
I find it hard to understand how does the fact that d is not part of the equation effects the basis. 

Thanks in advance.
$W$ is the kernel of a non zero linear form. Therefore the dimension of $W$ as a subspace of $\Bbb{R}^4$ is $4-1=3$. So it suffices to find three independent vectors belonging to $W$

$(0,0,0,1)$ is an obvious element of $W$

$(1,-1,0,0)$ and $(0,-1,1,0)$ are two other vectors.

The three vectors are independent therefore a basis.
Note that in the definition of $W$ you've given, $a+b+c = 0$ is best understood as a constraint on the 4-tuple $(a,b,c,d)$.

That is, there aren't any explicit constraints on $d$, so conceivably it can be whatever it wants.

Usually, linear algebra deals with real numbers, so it's my guess that $d\in R$, but that's not in the definition you've given, and so is just a guess on my part.