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  "body": "\n## Example\n\nLet $\\theta \\in \\R$.\nThe *rotation matrix* of angle $\\theta$ is\n$$R(\\theta) = \\begin{pmatrix} \\cos\\theta \u0026 -\\sin\\theta \\\\ \\sin\\theta \u0026 \\cos\\theta \\end{pmatrix}.$$\nThe map $x \\mapsto R(\\theta) x$ rotates the plane $\\R^2$ about the origin by the angle $\\theta$, counterclockwise.\nIt is an [[orthogonal matrix|0W02QESPJQ]] with determinant $\\cos^2\\theta + \\sin^2\\theta = 1$, so it is an element of $SO(2)$.\n\n## Remarks\n\nRotations add their angles: $R(\\alpha) R(\\beta) = R(\\alpha + \\beta)$.\nSo $R(\\theta)^{-1} = R(-\\theta) = R(\\theta)^T$.\nFor $\\theta$ not a multiple of $\\pi$, $R(\\theta)$ has no real eigenvalue; its complex eigenvalues are $e^{\\pm i\\theta}$.\nEvery element of $SO(2)$ is $R(\\theta)$ for some $\\theta$.\n\n## Explanations\n\n### L1 human\n\nThe rotation matrix sends $(1, 0)$ to $(\\cos\\theta, \\sin\\theta)$ and $(0, 1)$ to $(-\\sin\\theta, \\cos\\theta)$.\nThese are the first and the second column.\nFor $\\theta = 90$ degrees, it is $\\begin{pmatrix} 0 \u0026 -1 \\\\ 1 \u0026 0 \\end{pmatrix}$.\n",
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