[[Bilinear form]]
# Dual basis

Let $(V, \langle \cdot,\cdot  \rangle)$ be a [[Bilinear form#^nondegenerate]] finite-dimensional [[quadratic space]].
Given a [[vector basis|basis]] $\{ v_{i} \}_{i=1}^n$, the **dual basis** is a unique basis defined by #m/def/linalg 
$$
\begin{align*}
\langle v_{i}', v_{j} \rangle= \delta_{ij}
\end{align*}
$$


> [!missing]- Proof of existence and uniqueness
> #missing/proof


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