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  "body": "\n## Summary\n\nA system of linear integral equations has several unknown functions $\\varphi_1, \\ldots, \\varphi_n$, and each equation involves integrals of all of them.\nIn vector notation it is a single [[linear integral equation|AYZCKG34QK]] whose kernel is an $n \\times n$ [[matrix|0VD9ZSWWJG]] of functions.\nMuch of the theory for one equation, such as [[Fredholm theory|242TFD41KZ]], carries over.\nThe reason is that a system on an interval can be rewritten as one equation on a longer interval.\nSystems arise from coupled physical fields, from boundary integral methods for vector problems, and from the factorization of matrix symbols.\n\n## Explanations\n\n### L1 human\n\nSometimes one unknown function is not enough: two populations, two components of a displacement, two layers of a medium.\nEach unknown then satisfies an equation that also contains integrals of the others.\nStacking the unknowns into a vector turns the system into one equation with a matrix kernel.\nFor example, on $[0, 1]$ the pair $\\varphi_1(x) = x + \\frac{1}{2}$, $\\varphi_2(x) = x$ solves $\\varphi_1(x) - \\int_0^1 \\varphi_2(t)\\, dt = x$ and $\\varphi_2(x) - x \\int_0^1 \\varphi_1(t)\\, dt = 0$.\n",
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