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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $B$ be a [[bilinear form|0SQZMXB6GP]] on $V$.\nA family $(v_i)_{i \\in I}$ of vectors of $V$ is *orthogonal* for $B$ when\n$$B(v_i, v_j) = 0 \\qquad \\text{for all } i, j \\in I \\text{ with } i \\ne j.$$\n\n## Remarks\n\nFor an inner product, this is the usual notion of an orthogonal family.\nFor a general $B$, the relation $B(u, v) = 0$ need not be symmetric in $u$ and $v$; it is symmetric for symmetric and alternating forms.\nIn a basis that is orthogonal for $B$, the matrix of $B$ is diagonal.\nUnlike in an inner product space, a nonzero vector can be orthogonal to itself: for $B(x, y) = x_1 y_1 - x_2 y_2$, take $v = (1, 1)$.\n\n## Explanations\n\n### L1 human\n\nOrthogonality makes sense for every bilinear form, not only for inner products.\nTwo vectors are orthogonal when the form gives zero.\nA basis of pairwise orthogonal vectors makes the matrix of the form diagonal.\n",
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