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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.\nA *quadratic form* on $V$ is a map $Q \\colon V \\to F$ such that:\n\n1. $Q(av) = a^2 Q(v)$ for all $a \\in F$ and $v \\in V$.\n2. The map $(u, v) \\mapsto Q(u + v) - Q(u) - Q(v)$ is a [[bilinear form|0SQZMXB6GP]] on $V$.\n\n## Remarks\n\nOn $F^n$, the quadratic forms are the maps $Q(x) = \\sum_{i,j} a_{ij} x_i x_j$, that is, homogeneous polynomials of degree $2$.\nIn matrix notation, $Q(x) = x^T A x$ for an $n \\times n$ matrix $A$.\nIf the characteristic of $F$ is not $2$, the matrix $A$ can be chosen symmetric, and then it is unique.\nFor example, $Q(x, y) = x^2 + 3xy + 2y^2$ is a quadratic form on $\\R^2$.\n\n## Explanations\n\n### L1 human\n\nA quadratic form is a polynomial in which every term has degree two, such as $x^2 - 4xy + y^2$.\nIt is the simplest kind of function after linear functions.\nThe squared length $x_1^2 + \\cdots + x_n^2$ is the main example.\n",
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