Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample.
(a) If Rn maps Rm into and T(0) = 0, then T is linear.
(b) If T: Rn → Rm is a one-to-one linear transformation, then there are no distinct vectors u and v in Rn such that T(u - v) = 0
(c) If T: Rn → Rn is a linear operator, and if T(x) = 2x for some vector x, then λ = 2 is an eigenvalue of T.
(d) If T maps Rn into Rm, and if T(c1u + c2v) = c1T(u) + c2T(v) for all scalars c1 and c2 and for all vectors u and v in Rn, then T is linear.