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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 subspace of $V$.\nThen\n$$\\dim W + \\dim W^0 = \\dim V,$$\nwhere $W^0 \\subseteq V^*$ is the [[annihilator|0ASHNYF7AG]] of $W$.\n\n## Remarks\n\nA proof extends a basis $b_1, \\dots, b_k$ of $W$ to a basis $b_1, \\dots, b_n$ of $V$.\nThen the dual basis forms $b_{k+1}^*, \\dots, b_n^*$ are a basis of $W^0$.\nAnother proof uses the isomorphism $W^0 \\cong (V / W)^*$.\nWith the double dual isomorphism, it gives $(W^0)^0 = W$ in finite dimensions.\n\n## Explanations\n\n### L1 human\n\nThe more directions a subspace has, the fewer independent linear forms vanish on it.\nIn $\\R^3$, a line is killed by a plane of forms, and a plane is killed by a line of forms.\nThe two dimensions always add up to the dimension of the space.\n",
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