9 Suppose vi vn is a basis of V and 1Yn is the correspo

9 Suppose vi . . . . . vn is a basis of V and 1.····Yn is the corresponding dual basis of V. Suppose V. Prove that

Solution

Note v₁,,v_n is a basis of V and ₁,,_n is the corresponding dual basis of V, we have

((v₁)₁++(v_n)_n)(v₁)=(v₁).

Similarly, we also have

((v₁)₁++(v_n)_n)(v_i)=(v_i).

Hence

=(v₁)₁++(v_n)_n,

as they coincide at a basis of V.