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  "body": "\n## Summary\n\nA projection method approximates the solution of an operator equation by an element of a finite-dimensional subspace.\nThe main case is an [[integral equation of the second kind|8KJCN49V64]].\nOne chooses subspaces $X_n$ and projections $P_n$ onto them, and asks that the equation holds after applying $P_n$.\nThis gives a finite linear system for the coefficients of the approximation.\nThe [[Galerkin method|Y5B0PEBKE8]] (orthogonal projection) and the collocation method (interpolation) are the two main cases.\n\n## Explanations\n\n### L1 human\n\nAn integral equation asks for a whole function, which is infinitely many numbers.\nA projection method looks only among simple functions, for instance polynomials of degree at most $n$ or piecewise linear functions on a grid.\nIt cannot satisfy the equation exactly, so it asks that the error is invisible to the projection: zero at chosen points (collocation) or orthogonal to the chosen functions (Galerkin).\nFor example, for $\\varphi(x) - \\int_0^1 x t \\, \\varphi(t) \\, dt = x$ on $[0, 1]$ with $X_1$ the multiples of $x$, the Galerkin method gives $\\varphi = \\frac{3}{2} x$, which here is the exact solution.\n",
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