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  "body": "\n## Statement\n\nLet $V$ be a vector space of functions (for example, of sequences), and let $A$ be a linear operator on $V$.\nLet $p(A) = c_0 A^k + c_1 A^{k-1} + \\dots + c_k$ be a polynomial in $A$ with $c_0 \\ne 0$ and $c_k \\ne 0$.\nLet $g \\in V$, and let $W$ be the set of all $f \\in V$ with $p(A) f = 0$.\nSuppose $f_0 \\in V$ satisfies $p(A) f_0 = g$.\nThen every $f \\in V$ with $p(A) f = g$ can be written as $f = f_0 + f_1$ with $f_1 \\in W$.\n\n## Remarks\n\nProof: the operator $p(A)$ is linear, so $p(A)(f - f_0) = g - g = 0$. Put $f_1 = f - f_0$.\nThe set $W$ is the [[kernel|0S23E6YE83]] of $p(A)$, and so a subspace of $V$.\nThe conditions $c_0 \\ne 0$ and $c_k \\ne 0$ are not used in this argument.\nExample: for $u_{n+1} - 2u_n = 1$, the constant $-1$ is one solution, and $W$ is spanned by $2^n$.\n",
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