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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a vector space over $F$ of finite [[dimension|03GSZ8RTZJ]] $n$.\nLet $v_1, \\dots, v_n \\in V$ be $n$ vectors.\nIf $v_1, \\dots, v_n$ are [[linearly independent|0CRMGPAH83]], then they form a [[basis|0KCQ9Y3QKA]] of $V$.\nIf $v_1, \\dots, v_n$ span $V$, then they form a basis of $V$.\n\n## Remarks\n\nThus, for exactly $\\dim V$ vectors, independence and spanning are equivalent.\nFor example, three vectors of $\\R^3$ form a basis if and only if the $3 \\times 3$ matrix with these columns has nonzero determinant.\nMore generally, a family $v_1, \\dots, v_k$ is linearly independent if and only if its span has dimension $k$.\n\n## Explanations\n\n### L1 human\n\nWhen we have exactly as many vectors as the dimension, we only need to check one of the two basis conditions.\nIndependence gives spanning for free, and spanning gives independence for free.\nThis halves the work of checking a basis.\n",
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