In each case either prove the statement or give an example in which it is false. Throughout, let T: V → W be a linear transformation where V and W are finite dimensional.
(a) If V = W, then ker T ⊂ im T.
(b) If dim V = 5, dim W = 3, and dim(ker T) = 2, then T is onto.
(c) If dim V = 5 and dim W = 4, then ker T ≠ (0).
(d) If ker T = V, then W = {0}.
(e) If W = (0), then ker T = V.
(f) If W = V, and im T ⊂ ker T, then T = 0.
(g) If (el, e2, e3) is a basis of V and T(e1) = 0 = T(e2), then dim(im T) < 1.
(h) If dim(ker T) < dim W, then dim W > 1/2 dim V.
(i) If T is one-to-one, then dim V < dim W.
(j) If dim V < dim W, then T is one-to-one.
(k) If T is onto, then dim V > dim IV.
(1) If dim V > dim W, then T is onto.
(m) If T: V → W is linear and [T(v1),..., T(vk)} is independent, then (v1,..., vk) is independent.
(n) If T: T → W is linear and {v1,..., v2} spans V, then {T(v1),..., T(vk)} spans W.