I am given 4 vectors that span a subspace S of R^4.

w1 = (1,0,2,1)^T

w2= (1,1,2,0)^T

w3= (2,1,4,1)^T

w4 = (3,2,6,1)^T

I need to find the basis and dimension of S.

Here is what I get for my reduced row echelon form:

1 0 1 1

0 1 1 2

0 0 0 0

0 0 0 0

then i set the lead variables equal to the free variables and I get

x1 = -x3 - x4

x2 = -x3 - 2x4

so i think the basis would be {(-1,-1,1,0)^T,(-1,-2,0,1)^T}

but the answer says it is {(1, 0, 2, 1)^T, (0,1,1,2)^T)}

i know there can be more than one basis, is she just using that basis because it is w1 and w2? is my basis also a correct answer? thanks a lot.