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1) Let V be a finite dimensional vector space over R

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1) Let V be a finite dimensional vector space over R. Define

   V^© = {? Î F(V, R) | ? is a linear transformation}

(a) Prove that V^© is a subspace of F(V, R).

(b) Suppose B = {v₁,.., v_n} is a basis for V. Define ?₁,…, ?_n Î F(V, R) by

                      ?_i(a₁v₁+ … + a_nv_n) = a_i  for i = 1,.., n

Show that ?_i Î V^© for i = 1,. .., n and that the set B^© = {?₁,. .., ?_n} is a basis for V^©.

(c) Find and invertible linear transformation V ® (V^©)^©. (Don’t forget to prove that it is invertible!)

(d) Consider a linear transformation T: V ® W, where W is another finite-dimensional vector space over R. Define T^© : W^© ® V^© by

                                T^© (?) = ?oT

Prove that T^© is a linear transformation.

(e) In the setup of the previous part, suppose W = V so that T: V ® V and T^© : V^© ® V^©. Let M be the matrix representation of T with respect to an ordered basis B of V, and let M^© be the matrix representation of T^© with respect to the ordered basis B^© an defined in part (b).

Express M^© in terms of M.

(f) Let v₁,…, v_n Î V. Define T: V^© ® Rⁿ by

                            T(?) = (?(v₁),…, ?(v_n))

Prove that T is a linear transformation.

(g) Prove that the linear transformation T defined in part (f) is injective if and only if {v₁,.., v_n} spans V.

(h) ) Prove that the linear transformation T defined in part (f) is injective if and only if {v₁,.., v_n} is linearly independent.

(i) (An example)

Take V = P_n, the space of polynomials of degree at most n with real coefficients. Show that for any a Î R, the function ev_a : V ® R given by evaluation at a:

                                   Ev_a(f) = f(a)

Is an element of V^©.