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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.\nThen the [[space of linear maps|0X9MBPP2Z3]] $\\Hom_F(V, W)$ is finite-dimensional, and\n$$\\dim \\Hom_F(V, W) = \\dim V \\cdot \\dim W.$$\n\n## Remarks\n\nWith bases of $V$ and $W$, the map that sends $T$ to its matrix is an isomorphism $\\Hom_F(V, W) \\cong M_{m \\times n}(F)$, with $n = \\dim V$ and $m = \\dim W$.\nThe matrices $E_{ij}$ form a basis of $M_{m \\times n}(F)$, so its dimension is $mn$.\nIn particular, the [[dual space|07JBSZE0V4]] of $V$ has dimension $\\dim V$.\n\n## Explanations\n\n### L1 human\n\nA linear map from an $n$-dimensional space to an $m$-dimensional space is an $m \\times n$ matrix.\nA matrix of this size has $mn$ free entries.\nSo the space of these maps has dimension $mn$.\n",
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