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  "body": "\n## Statement\n\nLet $a \u003c b$, let $m \\ge 1$, and let $a_1, \\dots, a_m, b_1, \\dots, b_m \\in L^2[a, b]$.\nLet $K(x, y) = \\sum_{i=1}^{m} a_i(x) b_i(y)$ be the corresponding [[degenerate kernel|6Y1RY5767R]], let $\\lambda \\in \\C$ and $f \\in L^2[a, b]$.\nLet $A$ be the $m \\times m$ matrix and $F \\in \\C^m$ the vector with entries\n$$A_{ij} = \\int_a^b b_i(y) \\, a_j(y) \\, dy, \\qquad F_i = \\int_a^b b_i(y) \\, f(y) \\, dy.$$\nConsider the [[Fredholm equation of the second kind|DEYYNVCV9J]]\n$$\\varphi(x) - \\lambda \\int_a^b K(x, y) \\varphi(y) \\, dy = f(x).$$\nIts solutions $\\varphi \\in L^2[a, b]$ are exactly the functions $\\varphi = f + \\lambda \\sum_{i=1}^{m} c_i a_i$, where $c \\in \\C^m$ solves $(I_m - \\lambda A) c = F$.\nDistinct solutions $c$ give distinct functions $\\varphi$.\nIn particular, if $\\det(I_m - \\lambda A) \\ne 0$, the integral equation has exactly one solution for every $f$.\n\n## Remarks\n\nIf $\\varphi$ is a solution, put $c_i = \\int_a^b b_i(y) \\varphi(y) \\, dy$.\nThe equation says $\\varphi = f + \\lambda \\sum_i c_i a_i$, and multiplying this by $b_i$ and integrating gives $c_i = F_i + \\lambda \\sum_j A_{ij} c_j$.\nConversely, if $c$ solves the system and $\\varphi = f + \\lambda \\sum_i c_i a_i$, the same computation shows $\\int_a^b b_i \\varphi = c_i$, hence $\\lambda \\int_a^b K(\\cdot, y) \\varphi(y) \\, dy = \\lambda \\sum_i c_i a_i = \\varphi - f$.\nThis also recovers $c$ from $\\varphi$, which gives the injectivity.\n\nExample: on $[0, 1]$ with $K(x, y) = xy$, so $m = 1$ and $a_1(x) = b_1(x) = x$, one gets $A = (1/3)$.\nFor $\\lambda = 1$ and $f(x) = x$, $F = 1/3$, so $c = (1/3)/(2/3) = 1/2$ and $\\varphi(x) = 3x/2$.\nFor $\\lambda = 3$ the system is $0 \\cdot c = F$, solvable only when $F = \\int_0^1 y f(y) \\, dy = 0$.\n\nThe polynomial $\\det(I_m - \\lambda A)$ is the [[determinant|00KDZRW4C0]] that governs the problem; it equals the Fredholm determinant of $K$ when $K$ is continuous.\n\n## Explanations\n\n### L1 human\n\nWhen the kernel is a finite sum of products of a function of $x$ and a function of $y$, the integral equation hides only finitely many unknowns.\nThese unknowns are the numbers $\\int b_i \\varphi$, and they satisfy an ordinary square linear system.\nSolve the system, and the formula gives the solution of the integral equation.\n\n### L2 human\n\nDegenerate kernels are the base case of Fredholm theory.\nEvery square-integrable kernel can be approximated by degenerate kernels, and the general theorems are proved by reducing to this case plus a small remainder that a Neumann series handles.\nThe same reduction is a practical numerical method: expand the kernel in a few separable terms and solve a small linear system.\n",
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