Given n ≥ 2, let C2, C3,... , Cn be n - 1 columns in Rn. Define T: Rn → R by
T(X) = det([X C2 C3 ... Cn]) where [X C2 C3 ... Cn) is the n × n matrix with columns X, C2, C3,..., Cn. Show that T is a linear transformation; that is:
(a) Show that T(aX) = aT(X) for all X in Rn and all real numbers a.
(b) Show that T(X + Y) = T(X) + T(Y) for all X and Fin Rn.