Basis/Dual basis/Tautological lemma/Fact

Let ${}V$ be a finite-dimensional ${}K$ -vector space, endowed with a basis ${}v_{1},\ldots ,v_{n}$ , and the corresponding dual basis

{}v_{1}^{*},\ldots ,v_{n}^{*}\in {V}^{*}\,.

Then, for every vector

${}v\in V$ , the equality

{}v=\sum _{i=1}^{n}v_{i}^{*}(v)v_{i}\,.

holds.

The linear forms ${}v_{i}^{*}$ yield the scalars (coordinates) of a vector with respect to a basis.