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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.\nIf $\\dim W = \\dim V$, then $W = V$.\n\n## Remarks\n\nA basis of $W$ is a linearly independent family of $\\dim V$ vectors in $V$.\nSo it is a basis of $V$, see [[linearly independent set of the right size is a basis|096V7H13H9]].\nThe statement fails for infinite-dimensional spaces.\nIn $F[x]$, the subspace of polynomials with $p(0) = 0$ has the same dimension as $F[x]$, but it is not equal to $F[x]$.\n\n## Explanations\n\n### L1 human\n\nIf a subspace is as large as the whole space, measured by dimension, then it is the whole space.\nFor example, a two-dimensional subspace of the plane is the plane.\nThis lets us prove that two subspaces are equal by an inclusion and a count.\n",
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