Let V be an n-dimensional space and suppose that Rn. Fix a basis B for V

Question:

Let V be an n-dimensional space and suppose that ∈ Rn. Fix a basis B for V and consider the map h : V → R given → ∙ RepB() by the dot product.
(a) Show that this map is linear.
(b) Show that for any linear map g: V → R there is an ∈ Rn such that g = h.
(c) In the prior item we fixed the basis and varied the to get all possible linear maps. Can we get all possible linear maps by fixing an and varying the basis?