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  "body": "\n## Definition\n\nLet $R$ be a commutative ring, let $n \\ge 2$, let $i \\ne j$ be indices in $\\{1, \\dots, n\\}$ and let $c \\in R$.\nLet $E_{ij}$ be the $n \\times n$ matrix with entry $1$ in position $(i, j)$ and $0$ elsewhere.\nThe *transvection matrix* is\n$$T_{ij}(c) = I_n + c E_{ij}.$$\n\n## Remarks\n\nThe product $T_{ij}(c) A$ adds $c$ times row $j$ of $A$ to row $i$; this is an [[elementary matrix|01REXW3WWE]] of the row-addition type.\nIts determinant is $1$, and its inverse is $T_{ij}(-c)$.\nOver a field, the transvections generate the [[special linear group|0V9PMN0RBH]].\nIn the plane, $\\begin{pmatrix} 1 \u0026 c \\\\ 0 \u0026 1 \\end{pmatrix}$ is a horizontal shear.\n\n## Explanations\n\n### L1 human\n\nA transvection slides each point parallel to one axis, by an amount that depends on another coordinate.\nA deck of cards that we push sideways is a picture of a shear.\nAreas and volumes do not change, which matches the determinant $1$.\n",
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