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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ and $W$ be vector spaces over $F$.\nThe *space of linear maps* $\\Hom_F(V, W)$, also written $L(V, W)$, is the set of all [[linear maps|0NRYHJPQG2]] $V \\to W$. It has the operations\n$$(S + T)(v) = S(v) + T(v), \\qquad (aT)(v) = a\\, T(v)$$\nfor $S, T \\in \\Hom_F(V, W)$, $a \\in F$ and $v \\in V$.\nIt is a vector space over $F$, with the zero map as zero vector.\n\n## Remarks\n\nFor $V = F^n$ and $W = F^m$, the matrix map is an isomorphism $\\Hom_F(F^n, F^m) \\cong M_{m \\times n}(F)$.\nThe [[dimension of the space of linear maps|0Z9N1ER0W0]] is $\\dim V \\cdot \\dim W$ in finite dimensions.\nFor $W = F$, it is the [[dual space|07JBSZE0V4]] of $V$, and for $W = V$, it is the space of [[endomorphisms|0J7H0YBFDJ]] of $V$.\n\n## Explanations\n\n### L1 human\n\nLinear maps can be added and scaled, value by value, and the result is again linear.\nSo the linear maps from $V$ to $W$ form a vector space themselves.\nIn coordinates, this is the vector space of matrices of a fixed size.\n",
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