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  "body": "\n## Definition\n\nLet $F$ be a field, and let $V$ and $W$ be vector spaces over $F$.\nLet $L$ be the vector space over $F$ with basis the set $V \\times W$; write $[v, w]$ for the basis vector of $(v, w)$.\nLet $N$ be the subspace of $L$ spanned by all elements of these forms, for $v, v' \\in V$, $w, w' \\in W$ and $a \\in F$:\n$$[v + v', w] - [v, w] - [v', w], \\qquad [v, w + w'] - [v, w] - [v, w'],$$\n$$[av, w] - a[v, w], \\qquad [v, aw] - a[v, w].$$\nThe *tensor product* of $V$ and $W$ is the quotient space $V \\otimes W = L / N$.\nFor $v \\in V$ and $w \\in W$, the class of $[v, w]$ is written $v \\otimes w$.\n\n## Remarks\n\nThe map $(v, w) \\mapsto v \\otimes w$ is bilinear, and the vectors $v \\otimes w$ span $V \\otimes W$.\nNot every vector of $V \\otimes W$ is of the form $v \\otimes w$; for example, $e_1 \\otimes e_1 + e_2 \\otimes e_2$ in $F^2 \\otimes F^2$ is not.\nThe tensor product is characterised by its [[universal property|0YDRV01R8F]].\nFor finite-dimensional spaces, $V^* \\otimes W$ is isomorphic to the space of linear maps from $V$ to $W$.\n\n## Explanations\n\n### L1 human\n\nThe tensor product combines two vector spaces into one space of formal products $v \\otimes w$.\nThese products follow the rules of bilinearity and nothing more.\nIt turns bilinear maps into linear maps, which are easier to study.\n",
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