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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.\nLet $B = (b_1, \\dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.\nThe *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\n$$b_i^*(b_j) = \\delta_{ij} \\qquad (1 \\le i, j \\le n),$$\nwhere $\\delta_{ij} = 1$ if $i = j$ and $\\delta_{ij} = 0$ if $i \\ne j$.\n\n## Remarks\n\nThe dual basis is a basis of $V^*$.\nThe form $b_i^*$ returns coordinate $i$: $b_i^*(a_1 b_1 + \\cdots + a_n b_n) = a_i$.\nEach $f \\in V^*$ is $f = \\sum_i f(b_i)\\, b_i^*$.\nFor the standard basis of $F^n$, $e_i^*$ is the map $x \\mapsto x_i$.\n\n## Explanations\n\n### L1 human\n\nEach vector of the dual basis reads off one coordinate.\nThe form $b_1^*$ gives the first coordinate in the basis $B$, the form $b_2^*$ gives the second, and so on.\nThe condition $b_i^*(b_j) = \\delta_{ij}$ says exactly this.\n",
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