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T[x y]^T = [0 y^2]^T

A: If X = [0 1]^T then T(2X) = [0 4]^T while 2T(X) = [0 2]^T

I don't quite understand this solution. What are we trying to accomplish here? So, since T(2X) = [0 4]^T while 2T(X) = [0 2]^T do not yeild the same answer, then it's not linear? If the answer was T(2X) = [0 4]^T while 2T(X) = [0 4]^T, then it would be linear?