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  "body": "\n## Definition\n\nLet $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$.\nAn *orthonormal basis* of $V$ is a [[basis|0KCQ9Y3QKA]] $(e_1, \\dots, e_n)$ of $V$ that is an [[orthonormal family|0RW9TNNT87]].\n\n## Remarks\n\nAn orthonormal family of $\\dim V$ vectors is an orthonormal basis.\nIn an orthonormal basis, coordinates are inner products: $v = \\sum_i \\langle v, e_i \\rangle e_i$.\nThe vectors $\\frac{1}{\\sqrt 2}(1, 1)$ and $\\frac{1}{\\sqrt 2}(1, -1)$ form an orthonormal basis of $\\R^2$.\nIn infinite-dimensional Hilbert spaces, the term means a maximal orthonormal family, which is in general not a basis in the algebraic sense.\n\n## Explanations\n\n### L1 human\n\nAn orthonormal basis is a set of perpendicular unit vectors that spans the space.\nEach coordinate of a vector is its inner product with one basis vector.\nNo linear system needs to be solved to find coordinates.\n",
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