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Homework answers / question archive / 1) Let V be a finite dimensional vector space over R

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

Math

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,.., vn} is a basis for V. Define ?1,…, ?n Î F(V, R) by

                      ?i(a1v1+ … + anvn) = ai  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,…, vn Î V. Define T: V© ® Rn by

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

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,.., vn} spans V.

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

(i) (An example)

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

                                   Eva(f) = f(a)

Is an element of V©.

 

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