Let V_n be the set of polynomials with real coefficients of degree at most n, that is a_n*x^n+...+a_0 ∈ V_n.

a) Prove that {x^n, x^n-1,..., 1} is linearly independent.

b) Show that span {x^n, x^n-1,..., 1} = V_n.

c) Deduce that V_n is a vector space with basis {x^n, x^n-1,...,1}.

d) Find the dimension of V_n.

e) Show that the derivative is a linear map; d/dx: V_n --> V_n-1.

f) Find the matrix associated to d/dx with respect to the bases for V_n and V_n-1 from part (c).

( _ denotes subscripts while ^ denotes superscripts)

I'm not sure where to start with this one so if you guys can steer me in the right direction, it would be great. Thanks.