Indicate whether each statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. In each part, V and W are vector spaces.
(a) If T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for vectors v1 and v2 in Land all scalars c1 and c2, then T is a linear transformation.
(b) If v is a nonzero vector in V, then there is exactly one linear transformation
T: V → W such that T(-v) = -T(v).
(c) There is exactly one linear transformation T: V → W for which T(u + v) = T(u - v) for all vectors u and v in V.
(d) If v0 is a nonzero vector in V, then the formula T(v) = v0 + v defines a linear operator on V.