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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$, and let $(e_1, \\dots, e_n)$ be an [[orthonormal basis|0S9GGTR0EX]] of $V$.\nThen, for each $v \\in V$,\n$$v = \\sum_{i=1}^{n} \\langle v, e_i \\rangle\\, e_i.$$\nSo the coordinates of $v$ in this basis are the inner products $\\langle v, e_i \\rangle$.\n\n## Remarks\n\nIf $v = \\sum_j a_j e_j$, then $\\langle v, e_i \\rangle = \\sum_j a_j \\langle e_j, e_i \\rangle = a_i$.\nFor the basis $\\frac{1}{\\sqrt 2}(1, 1)$, $\\frac{1}{\\sqrt 2}(1, -1)$ of $\\R^2$, the vector $(3, 1)$ has the coordinates $4/\\sqrt 2$ and $2/\\sqrt 2$.\nWith the convention of linearity in the second argument, the coefficient is written $\\langle e_i, v \\rangle$.\n\n## Explanations\n\n### L1 human\n\nIn an orthonormal basis, we find each coordinate of a vector with one inner product.\nThis is much easier than solving a linear system.\nIt is the finite-dimensional form of a Fourier expansion.\n",
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