Question

why would the following solution space have no basis:

2x_1+x_2+3x_3=0
x_1 + 0 +5x_3=0
0 + x_2+x_3=0

I know it reduces to
1 0 0
0 1 0
0 0 1

but why does that mean it has no basis? I would think that a 3x3 identity matrix of 3 vectors i, j ,k can span R^3