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Transformation induced dual basis [math-000D]
Manifolds and Differential Geometry

Let e1,e2,…,en∈Ve_1, e_2, \dots, e_n \in V be a basis of vector space VV, and let e1,…,en∈V∗e^1, \dots, e^n \in V^* be a basis of dual space V∗V^*. Now if eˉi=C  ikek\bar{e}_i = C^k_{\ \ i} e_k is another basis of VV, then there is an induced basis eˉi=(C−1)  kiek\bar{e}^i = (C^{-1})^i_{\ \ k} e^k for dual space V∗V^*.

figure tex2120

Proof

By Kronecker-delta 1=δ ii1 = \delta^i_{\ i}

1=δ ii=eˉieˉi=C  ikekeˉi1 = \delta^{i}_{\ i} = \bar{e}_{i} \bar{e}^{i} = C^k_{\ \ i} e_{k} \bar{e}^{i}

can see that if eˉi=ek(C−1)ki\bar{e}^{i} = e^{k} (C^{-1})^i_k then the equality is hold. We can use Penrose notation to show the idea.

figure tex2121