How to find the basis for a set of vectors

[HappyN]

Homework Statement

The question states:
Consider the subset S of R^4 given by:
S={(2,3,-1,7), (1,0,1,3), (0,3,-3,1), (12,15,-3,29)}
i) Decide whether the vectors in S form a linearly independent set.
ii) Let V be the vector subspace of R^4 spanned by the vectors of S, i.e:
V=span{ (2,3,-1,7), (1,0,1,3), (o,3,-3,1), (12,15,-3,29)}
Find a basis for V and write down the dimension of V.

The Attempt at a Solution

For the first part of the question, I solved:
2a + b + 12d=0
3a+ 3c+15d=0
-a+b-3c-3d=0
7a+3b+c+29d=0
as a system of linear equations, I solved this by putting it in matrix form and got:
d=0, a= -1/2(b) = -c
and hence I concluded that the vectors do not form a linearly independent set.
For part ii) I first sifted the set of 4 vectors and got the following 3 vectors remaining: {(2,3,-1,7), (1,0,1,3), (0,3-3,1)}, I then tired to solve these vectors with respect to the natural basis (1,0,0,0), (0,1,0,0), (0,0,1,0) and (0,0,0,1) but ended up getting a contradiction in my answers - please tell me if I have been going along the right lines and whether my part i) is correct - any help would be greatly appreciated! thanks

 

[ojs]

Why would you try to solve the three vectors with respect to the natural basis? To form a basis you just need some vectors that span the subspace and are linearly independent. Aren't those three you got a good choice?