step one: convert all the matrices in S into vectors in R^{4}.

this turns S into span({(1,1,0,0),(2,2,0,0),(0,1,0,0),(1,2,0,0),(15, 1,0)}

it should be obvious we can eliminate (2,2,0,0) from this set without fear, since it's just 2(1,1,0,0).

also note that (1,2,0,0) = (1,1,0,0) + (0,1,0,0), so we don't need (1,2,0,0), either (it's a linear combination of the first two vectors).

that leaves just 3 vectors left (1,1,0,0), (0,1,0,0) and (1,5,1,0). is this a linearly independent set?