As part of my Computer Science degree, we're required to do a course on Linear Algebra and a few other things. The first set of lectures are on vector subspaces, but I have a question (it's part of an assignment I've pretty much completed, I just don't understand if/why this works):

W = {(x,y,z) | x + 3y - 2z = 0 } C R3 (is a subset of R3)

The problem is to show whether or not W is a vector subspace.

Vector Addition

v = (v1,v2,v3)
w = (w1,w2,w3)
v and w are in W.

(v1 + w1) + 3(v2 + w2) - 2(v3 + w3) =
(v1 + w1) + (3v2 + 3w2) - (2v3 + 2w3) =
(v1 + 3v2 - 2v3) + (w1 + 3w2 - 2w3) =
0 + 0 = 0

I've been told that this (along with a similar proof involving scalar multiplication) shows that W is a vector subspace, but why does it prove it?