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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]].\nThe *dual map* of $T$, also called its *transpose*, is the map\n$$T^* \\colon W^* \\to V^*, \\qquad T^*(f) = f \\circ T,$$\nbetween the [[dual spaces|07JBSZE0V4]].\n\n## Remarks\n\nThe dual map is linear.\nIt reverses the order of composition: $(T \\circ S)^* = S^* \\circ T^*$, and $(\\id_V)^* = \\id_{V^*}$.\nIn dual bases, the matrix of $T^*$ is the transpose of the matrix of $T$, see [[the matrix of the dual map|0ZPB3180AY]].\nThe kernel of $T^*$ is the [[annihilator|0ASHNYF7AG]] of the image of $T$.\n\n## Explanations\n\n### L1 human\n\nA linear form on $W$ can be pulled back to $V$: first apply $T$, then the form.\nThe dual map does this pullback.\nIt goes in the opposite direction, from $W^*$ to $V^*$.\n",
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