Exercise 7. Let V be a nitedimensional vector space over IF. The vector space V* : (V, F) is called the dual of V. An element of V* is thus a linear...

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Algebra 2.

MATH 236 in mcgill

Exercise 7. Let V be a finite—dimensional vector space over IF. The vector space V* :£(V, F) is called the dual of V. An element of V* is thus a linear map f : V —> IF, alsoknown as a functional. (i) Let {U1,. .. ,vn} be a basis of V. Show that, for each i E {1, . . . ,n}, there exists aunique ’03“ E V* satisfying1 if 2' : j vi(vj)6i'{0 ifz’7éj' Show that {01“, . . . , 11;} is a basis of V*. Deduce that V and V* are isomorphic.(ii) For each 1: E V, let evv : V* —> IF denote the evaluation at 1); that is, evu(f) : f (v)for each f E V*. Check that evv E V**, the dual of V*. Show that the evaluationmap ev : V —> V** is an isomorphism. (A brief comment. The isomorphism of V and V** is canonical, in the sense that it doesnot depend on a basis. The isomorphism of V and V*, on the other hand, is not canonical—it requires the choice of a basis. The notion of duality, and the canonical/non—canonicalisomorphisms exhibited above, appear in relation to several mathematical structures.)