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  "body": "\n## Definition\n\nLet $R$ be a ring and let $A = (a_{ij})$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] over $R$.\nThe *transpose* of $A$ is the $n \\times m$ matrix $A^T$ with entries\n$$(A^T)_{ij} = a_{ji}$$\nfor $1 \\le i \\le n$ and $1 \\le j \\le m$.\n\n## Remarks\n\nThe transpose of $\\begin{pmatrix} 1 \u0026 2 \u0026 3 \\\\ 4 \u0026 5 \u0026 6 \\end{pmatrix}$ is $\\begin{pmatrix} 1 \u0026 4 \\\\ 2 \u0026 5 \\\\ 3 \u0026 6 \\end{pmatrix}$.\nThe rules $(A^T)^T = A$, $(A + B)^T = A^T + B^T$ and $(cA)^T = c A^T$ hold.\nOver a commutative ring, $(AB)^T = B^T A^T$, see [[transpose of a product|09GK5PS824]].\nThe [[matrix of the dual map is the transpose|0ZPB3180AY]] of the matrix of the map.\n\n## Explanations\n\n### L1 human\n\nTo transpose a matrix, flip it over its main diagonal.\nRow $1$ becomes column $1$, row $2$ becomes column $2$, and so on.\nA $2 \\times 3$ matrix becomes a $3 \\times 2$ matrix.\n",
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