Find a basis for the solution space of the homogeneous system of equations.

Find a basis for the solution space of the homogeneous system of equations.

2x1 + x2 + x3 + x4 = 0,

3x1 + x2 - x3 + 2x4 = 0,

x1 + x2 + 3x3 = 0,

x1 - x2 - 7x3 + 2x4 = 0

I tried putting it in row-echelon form which gave me

[ 2 1 1 1 ]

[ 0 -1 -5 1 ]

[ 0 0 0 0 ]

[ 0 0 0 0 ]

let x2=0 , x3=1 which gives [-3 0 1 5]

x4 =5 , x1= -3

let x3=0, x2=x4=1 which gives [-1 1 0 1]

x1=-1

and dim 2..

but i have a feeling this may be wrong?

Thanks in advance