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.

Proof, Write another proof