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  "body": "\n## Statement\n\nLet $F$ be a field, and let $V$ and $W$ be vector spaces over $F$.\nLet $(e_i)_{i \\in I}$ be a [[basis|0KCQ9Y3QKA]] of $V$ and let $(f_j)_{j \\in J}$ be a basis of $W$.\nThen the family $(e_i \\otimes f_j)_{(i, j) \\in I \\times J}$ is a basis of the [[tensor product|0BF2DE2SYK]] $V \\otimes W$.\n\n## Remarks\n\nSo each vector of $V \\otimes W$ is uniquely $\\sum_{i,j} c_{ij}\\, e_i \\otimes f_j$ with only finitely many nonzero $c_{ij} \\in F$.\nFor $v = \\sum_i a_i e_i$ and $w = \\sum_j b_j f_j$, $v \\otimes w = \\sum_{i,j} a_i b_j\\, e_i \\otimes f_j$.\nThus the coefficients of a vector $v \\otimes w$ form a matrix of rank at most $1$.\n\n## Explanations\n\n### L1 human\n\nFrom a basis of each factor, we get a basis of the tensor product.\nIts vectors are all the products of a basis vector of $V$ and a basis vector of $W$.\nA vector of $V \\otimes W$ is like a matrix of coefficients.\n",
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