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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a [[subspace|00YC1D652K]] of $V$.\nThe *annihilator* of $W$ is the subset of the [[dual space|07JBSZE0V4]]\n$$W^0 = \\{f \\in V^* : f(w) = 0 \\text{ for all } w \\in W\\}.$$\n\n## Remarks\n\nThe annihilator $W^0$ is a subspace of $V^*$.\nIf $W_1 \\subseteq W_2$, then $W_2^0 \\subseteq W_1^0$.\nIn finite dimensions, $\\dim W + \\dim W^0 = \\dim V$, see [[the dimension of the annihilator|06Q1EFVH0Q]].\nThe annihilator of the line spanned by $(1, 1)$ in $F^2$ is spanned by the form $(x, y) \\mapsto x - y$.\n\n## Explanations\n\n### L1 human\n\nThe annihilator of a subspace collects the linear forms that are zero on all of it.\nIn $\\R^3$, the forms that vanish on a plane through the origin are the multiples of one form, whose kernel is that plane.\nA larger subspace has a smaller annihilator.\n",
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