Question

4. Bonus 8 pt total] This question is about dual vector spaces and their connection to the transpose If V is a vector space, then the vector space Hom(V, R), of linear maps from V to R is called the dual vector space of V and we denote it by V". a) Suppose ei Is a basis for V. Show or which we have I i=j (b) Show that the set e is linearly independent. (c) If V is finite dimensional, show that e" is a basis of V We call e the dual basis of V. (d) Show that if V is infinite dimensional, then the e" are not a basis. (e) Notice that given a map L : V → W we obtain a map W*-Hom(W,R) → Hom( V,R) = V* by sending h to hoL, we shall call this map L* the dual map of g. Show that if V and W are finite dimensional with basis ei and f respectively, that if the matrix for L with respect to these basis is M, then the matrix for L with respect to the basis f and e is Mt, the transpose of M

Answer 1

The answers to the questions (d) and (e) are shuffled. Please Note.
Fs cac -) and eudend* леса at ,几 an →长

: a bo is But Len L-O) J- to 2 ス+- basi →大 イ a > 나

12 しー(3) --)女 -)ん m 上?' Or) r-, r) 9っっっっっっっっ'^^

a n MI in a v an ーナ e ap Poce -う女

Sinc k is inti ke can cheose je k at he can C hoose l=o cs.tr a d 'chồ n a Cle re a ba Is