Suppose v1 vn is a basis of V and phi 1 phi n is the corr

Suppose v1.... , vn is a basis of V and phi 1. . . phi n is the corresponding dual basis of V\'. Suppose psi in 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.