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  "body": "\n## Statement\n\nLet $V$ and $W$ be finite-dimensional inner product spaces over $\\R$ or $\\C$, with [[orthonormal bases|0S9GGTR0EX]] $B$ and $C$.\nLet $T \\colon V \\to W$ be a linear map.\nThen the matrix of the [[adjoint|0R9NMSM1ZP]] is\n$$[T^*]_{B,C} = \\left([T]_{C,B}\\right)^*,$$\nwhere $^*$ on the right is the [[conjugate transpose|0DXKE10PHF]]. Here $[\\,\\cdot\\,]$ is the [[matrix of a linear map|0YQSGNYE3K]].\n\n## Remarks\n\nOver $\\R$, the matrix of $T^*$ is the transpose of the matrix of $T$.\nThe statement needs orthonormal bases; in other bases, the Gram matrices of the bases appear in the formula.\nThe proof compares entries: both are $\\langle T(b_j), c_i \\rangle$, up to conjugation.\n\n## Explanations\n\n### L1 human\n\nIn orthonormal coordinates, taking the adjoint of a map means taking the conjugate transpose of its matrix.\nThis makes the abstract adjoint easy to compute.\nIn other bases, the rule is more complicated.\n",
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