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  "body": "\n## Statement\n\nLet $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$.\nLet $B$ be a basis of $V$ and $C$ a basis of $W$, with [[dual bases|0ZD2CF6ZS0]] $B^*$ and $C^*$.\nLet $T \\colon V \\to W$ be a linear map with [[dual map|0AZ5ARP287]] $T^*$.\nThen\n$$[T^*]_{B^*, C^*} = \\left([T]_{C, B}\\right)^T,$$\nwhere $[\\,\\cdot\\,]$ is the [[matrix of a linear map|0YQSGNYE3K]] and the exponent $T$ is the [[transpose|09DXWYRD43]].\n\n## Remarks\n\nThe proof computes entry $(j, i)$ of $[T^*]$ as $T^*(c_i^*)(b_j) = c_i^*(T(b_j))$, which is entry $(i, j)$ of $[T]$.\nWith the [[annihilator|0ASHNYF7AG]], this gives a proof that [[row rank equals column rank|0TZAG60ZYT]].\n\n## Explanations\n\n### L1 human\n\nThe dual map is the linear algebra form of the transpose of a matrix.\nIn dual bases, its matrix is the transpose of the matrix of the original map.\nThis explains why rows and columns play symmetric roles.\n",
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