how to solve this, T is surjective but why can T* is injective?

In the rst two questions we will consider the following. Let V, W be vector spaces over a eld F and let T : V > W be a linear map- Dene T\" : [W,F) > ICU/,F) by (T*)(f) = f oT. This is a linear map (you do not need to prove this). 1. Prove the following. (a) If T is surjective then T* is injective. [b] if T has a left inverse, i.e.1 a linear map 3 : W > V such that SOTZidv, then T* is surjective. 2. Let o: = {o1,...,vn} be a nite ordered basis for V. For 1' = l,...,n, dene f3: : V 4; F by f(alvl + ' ' ' + anvil.) = i- rI'hen ,8 = {f13 . . . , n} is a basis for the vector space (V, F) (you don't need to prove this). Let T : V > V be a linear operator. Prove that [T15 = ([Tla)\". (the transpose of the matrix [T]Q.)