1. The problem statement, all variables and given/known data An affine transformation T : R^n --> R^m has the form T(x) = Ax + y, where x,y are vectors and A is an m x n matrix and y is in R^m. Show that T is not a linear transformation when b != 0. 2. Relevant equations There are couple of useful definitions, but I think this one will suffice : Definition : If T is a linear transformation, then T(0) = 0 and T(cU + dV) = cT(U) + dT(V), for all vectors U,V in the domain of T and all scalars c,d. 3. The attempt at a solution I figure I would show that the affine transformation does not satisfy the definition. Attempt : Let U,V be vectors, and c,d be scalers. Then the affine transformation has to satisfy T(cU + dV) ?= cT(U) + dT(V). since T(x) = Ax + y then T(cU + dV) = A(cU + dV) + y = cAU + dAV + y = cAU + dT(V) thus cAU + dT(V) != cT(U) + dT(V), which means that affine transformation is not a linear transformation. Would this suffice or Do I make some assumption that are not plausible. BTW, there was a theorem that said that If A is an m x n matrix, then the transformation x -> Ax has the properties A(U + V) = AU + aV and A(sU) = sA(U), where U,V are vectors in R^n and s is a scaler.