Dual basis - Knowlpedia

Let $(e_1,\ldots,e_n)$ be a basis of a finite-dimensional vector space $V$ . Its dual basis is the family $(e^1,\ldots,e^n)$ of linear functionals satisfying

e^i(e_j)=\delta_{ij},

where $\delta_{ij}$ is one if $i=j$ and zero otherwise. For $v=\sum_j v_j e_j$ , the functional $e^i$ returns the coordinate $v_i$ , so

v=\sum_i e^i(v)e_i.

Basis of the dual

Every linear functional $\ell$ equals $\sum_i\ell(e_i)e^i$ . Evaluating a proposed linear relation among the $e^i$ on each $e_j$ shows independence. Hence the family is a basis of the dual space. No inner product or orthogonality is required.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.