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  "body": "\n## Definition\n\nLet $V$ and $W$ be finite-dimensional inner product spaces over $\\R$ or $\\C$, and let $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]].\nThe *adjoint* of $T$ is the unique linear map $T^* \\colon W \\to V$ such that\n$$\\langle T(v), w \\rangle = \\langle v, T^*(w) \\rangle \\qquad \\text{for all } v \\in V \\text{ and } w \\in W.$$\n\n## Remarks\n\nThe existence and uniqueness follow from the [[Riesz representation theorem|04GYPXDXNH]].\nIt satisfies $(S + T)^* = S^* + T^*$, $(aT)^* = \\bar a T^*$, $(ST)^* = T^* S^*$ and $T^{**} = T$.\nIn orthonormal bases, the matrix of $T^*$ is the [[conjugate transpose|0DXKE10PHF]] of the matrix of $T$.\nIt satisfies $\\ker T^* = (\\im T)^\\perp$.\n\n## Explanations\n\n### L1 human\n\nThe adjoint lets us move a map from one side of an inner product to the other.\nFor real matrices with the dot product, the adjoint is the transpose: $(Ax) \\cdot y = x \\cdot (A^T y)$.\nFor complex matrices, it is the conjugate transpose.\n",
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