Find a basis from cartesian equations

[Tonig00]

Hello
I want to find an easier way to find vectors which are a basis of a vector space defined by cartesian equations. For eixample:
x1+2x2+x3+2x4+x5=0
-x1-2x2-x4+2x5=0
x1+2x2+3x3+4x4+7x5=0

So, first I reduce the system (applying Gauss reduction):
I use RREF(matrix of coefficients of the system)
rref([[1,2,1,2,1],[-1,-2,0,-1,2],[1,2,3,4,7]])
to obtain
[[1,2,0,1,-2],[0,0,1,1,3],[0,0,0,0,0]]

So we have two equations which means the subspace represented have 5-2 dimensions. So for a basis of this space we need three independent vectors.

Next I use solve() giving values to find three possible vectors:
solve([[x1+2*x2+x4-2*x5,x3+x4+3*x5,x2 = 1,x4 = 0,x5 = 0]],[x1,x2,x3,x4,x5])
which gives {[-2,1,0,0,0]}

So I will change values for x2=0, x4=1 and X5=0 which gives {[-1,0,-1,1,0]}

And third and last I change x2=0, X4=0 and X5=1 which gives {[2,0,-3,0,1]}

This is a very complicated way, to obtain the three vectors of the basis.
Is there a function to solve this directly?

Thanks for your help

Toni