## Dual Vector Space and Dual Basis, Some Equality

## Problem 282

Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$.

Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that

\[x=\sum_{i=1}^nv^i(x)v_i\]
for any vector $x\in V$.