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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ and $W$ be [[vector spaces|000AYPA9AJ]] over $F$.\nThe *product* $V \\times W$ is the set of pairs $(v, w)$ with $v \\in V$ and $w \\in W$, with the operations\n$$(v, w) + (v', w') = (v + v', w + w'), \\qquad a(v, w) = (av, aw)$$\nfor $v, v' \\in V$, $w, w' \\in W$ and $a \\in F$.\nIt is a vector space over $F$ with zero vector $(0, 0)$.\n\n## Remarks\n\nFor finitely many spaces, the product is also the [[external direct sum|0S7W6EG32V]], written $V \\oplus W$.\nThe [[dimension of a direct product|06HN47VPTH]] is $\\dim V + \\dim W$.\nFor example, $F^m \\times F^n$ is isomorphic to $F^{m+n}$.\nThe projections $(v, w) \\mapsto v$ and $(v, w) \\mapsto w$ are linear maps.\n\n## Explanations\n\n### L1 human\n\nThe product of two vector spaces has vectors that are pairs: one vector from each space.\nWe add pairs part by part and scale both parts.\nThe plane $\\R^2$ is the product of two copies of the line $\\R$.\n",
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