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  "body": "\n## Proof\n\n1. Let $m = \\dim V$ and $n = \\dim W$. By the definition of [[dimension|05G2G0P224]], $V$ has a [[basis|0VTQTPQGAV]] $e_1, \\dots, e_m$ and $W$ has a basis $f_1, \\dots, f_n$.\n2. By the [[basis of a tensor product|0SH5VT8QYA]], the family $(e_i \\otimes f_j)$ with $1 \\le i \\le m$ and $1 \\le j \\le n$ is a basis of $V \\otimes W$.\n3. This family has $mn$ indices. Its vectors are pairwise distinct, because a [[linearly independent|0EVHJFTG9Z]] family has no repeated vector. A repeat $x = y$ would give the nontrivial relation $1 \\cdot x - 1 \\cdot y = 0$.\n4. So $V \\otimes W$ has a basis of $mn$ vectors. Hence it is [[finite-dimensional|018W3V3TCT]], and $\\dim(V \\otimes W) = mn = \\dim V \\cdot \\dim W$.\n\n## Idea\n\nThe products $e_i \\otimes f_j$ of basis vectors form a basis of $V \\otimes W$. There is one such product for each pair $(i, j)$, which gives $\\dim V \\cdot \\dim W$ vectors.\n",
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