1. Set of Polynomials

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.

2. Hi!

(a) How many zeros can a polynomial with degree n != 0 have?
(b) definition of span
(c) definition of a basis
(d) definition of dimension
(e) that follows from rules how to calculate the derivative
(f) the (coordinates of the) images of the basis vectors form the columns of the associated matrix

That is a good exercise to get used to the concept of a vector space. You just have to follow definitions so there should not be any problems.

3. To show (a), an easy way is to suppose a polynomial is identically zero, and take its derivative until you get to a contradiction.

Otherwise you can use the Vandermonde determinant but that's a little more complicated.