Show that each of the following functions is a linear transformation.
(a) T: R2 → R2; T(x, y) = (x, -y) (reflection in the X axis)
(b) T: R3 → R3; T(x, y, z) = (x, y, -z) (reflection in the X-Y plane)
(c) T: C → C; T(z) = z (conjugation)
(d) T: Mmn → Mkl; T(A) = PAQ, P a k × m matrix, Q an n × I matrix, both fixed
(e) T: Mnn → Mnn T(A) = AT + A.
(f) T: Pn → R; T[p(x)] = p(0)
(g) T: Pn → R; T(r0 + r1x + . . . . . . . + rnxn) = rn
(h) T: Rn → R; T(X) = X • Z, Z a fixed vector in Rn
(i) T: Pn → Pn; T[p(x)] = p(x + 1)
(j) T: Rn → V; T[r1 ∙ ∙ ∙ rn]T = r1e1 + ∙ ∙ ∙ + rnen where {e1,..., en} is a fixed basis of V.
(k) T: V → R; T(r1e1 + ∙ ∙ ∙ + rnen) = r1 where {e1,..., en} is a fixed basis of V