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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a finite-dimensional vector space over $F$.\nThen the [[dual space|07JBSZE0V4]] $V^*$ is finite-dimensional, and\n$$\\dim V^* = \\dim V.$$\n\n## Remarks\n\nThe [[dual basis|0ZD2CF6ZS0]] of a basis of $V$ is a basis of $V^*$ with the same number of elements.\nThis also follows from $\\dim \\Hom_F(V, W) = \\dim V \\cdot \\dim W$ with $W = F$.\nThe isomorphism $V \\to V^*$ that sends $b_i$ to $b_i^*$ depends on the basis.\nFor infinite-dimensional $V$, the dimension of $V^*$ is strictly greater than the dimension of $V$.\n\n## Explanations\n\n### L1 human\n\nA linear form on an $n$-dimensional space is fixed by its $n$ values on a basis.\nThese $n$ values can be any numbers.\nSo the linear forms make up an $n$-dimensional space too.\n",
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