dual basis

Given a vector space $V$ with a basis $\beta =\{v_1,v_2,\dots ,v_n\}$, there exists a unique basis of the dual space $\{{\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_n\}$ such that

$$\begin{array}{rl}{\epsilon }_i:V& ⟶\mathbb{F}\\ v_j& ⟼\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}\end{array}$$

Where $v_j\in \beta$.

This is called the dual basis for the basis $\beta$

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References