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  "body": "\n## Definition\n\nLet $R$ be a commutative ring, and let $E$ be a linear recurrence with constant coefficients over $R$ of order $d$.\nThe space of solutions of $E$ is the set of all [[solutions|0ZV2YKKZ58]] of $E$.\nIt is a submodule of the $R$-module $R^{\\mathbb{N}}$ of all sequences in $R$, with termwise addition and scalar multiplication.\n\n## Remarks\n\nThe map $u \\mapsto (u_0, \\dots, u_{d-1})$ is an isomorphism of $R$-modules from the space of solutions to $R^d$. Over a field, the space of solutions has dimension $d$.\nFor the Fibonacci recurrence over $\\mathbb{R}$, the sequences $\\varphi^n$ and $\\psi^n$ form a basis, with $\\varphi$ and $\\psi$ the roots of $X^2 - X - 1$.\n\n## Explanations\n\n### L1 human\n\nIf two sequences solve the same linear recurrence, then so does their sum, and so does any multiple of one of them.\nSo the solutions form a vector space, and the first $d$ terms are coordinates on it.\n",
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