Dual basis

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 of $V$. The dual basis of $B$ is the family $(b_1^*, \dots, b_n^*)$ of elements of the dual space $V^*$. 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.