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  "body": "\n## Definition\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, and let $v_1, \\dots, v_k \\in V$.\nThe *Gram matrix* of $v_1, \\dots, v_k$ is the $k \\times k$ matrix $G$ with entries\n$$G_{ij} = \\langle v_i, v_j \\rangle \\qquad (1 \\le i, j \\le k),$$\nwhere $\\langle \\cdot, \\cdot \\rangle$ is the [[inner product|0NB0AWGJBW]].\n\n## Remarks\n\nThe Gram matrix is Hermitian and [[positive semidefinite|01MMVW8CKC]].\nIt is invertible if and only if $v_1, \\dots, v_k$ are linearly independent.\nFor real column vectors $v_i \\in \\R^n$ with the dot product, $G = A^T A$, where $A$ has the columns $v_1, \\dots, v_k$.\nIts determinant is the square of the $k$-dimensional volume of the parallelepiped spanned by the $v_i$.\n\n## Explanations\n\n### L1 human\n\nThe Gram matrix records all inner products of a list of vectors.\nIt contains all lengths and all angles between them.\nTwo lists with the same Gram matrix have the same shape, up to an isometry.\n",
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