---
tag: 0W6M3PSD08
kind: definition
name: Dual basis
msc: [15A04]
defines: 0ZD2CF6ZS0
uses: [0M856Q1AMZ, 0VTQTPQGAV]
sources:
  - url: https://en.wikipedia.org/wiki/Dual_basis
    locator: Finite-dimensional vector spaces
---

## Definition

Let $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.
Let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
The *dual basis* of $B$ is the family $(b_1^*, \dots, b_n^*)$ of elements of the [[dual space|07JBSZE0V4]] $V^*$. These elements are defined by
$$b_i^*(b_j) = \delta_{ij} \qquad (1 \le i, j \le n),$$
where $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \ne j$.

## Remarks

The dual basis is a basis of $V^*$.
The form $b_i^*$ returns coordinate $i$: $b_i^*(a_1 b_1 + \cdots + a_n b_n) = a_i$.
Each $f \in V^*$ is $f = \sum_i f(b_i)\, b_i^*$.
For the standard basis of $F^n$, $e_i^*$ is the map $x \mapsto x_i$.

## Explanations

### L1 human

Each vector of the dual basis reads off one coordinate.
The form $b_1^*$ gives the first coordinate in the basis $B$, the form $b_2^*$ gives the second, and so on.
The condition $b_i^*(b_j) = \delta_{ij}$ says exactly this.