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  "body": "\n## Statement\n\nLet $F$ be a field, and let $V$ and $W$ be finite-dimensional vector spaces over $F$.\nThen\n$$\\dim (V \\otimes W) = \\dim V \\cdot \\dim W,$$\nwhere $\\dim$ is the [[dimension|03GSZ8RTZJ]] over $F$.\n\n## Remarks\n\nIt follows from the [[basis of a tensor product|0SH5VT8QYA]], which has $\\dim V \\cdot \\dim W$ vectors.\nFor example, $F^m \\otimes F^n$ has dimension $mn$, like the space of $m \\times n$ matrices.\nCompare this with the direct sum: $\\dim (V \\oplus W) = \\dim V + \\dim W$.\n\n## Explanations\n\n### L1 human\n\nThe dimension of a tensor product is the product of the dimensions.\nThe dimension of a direct sum is their sum instead.\nSo $\\R^2 \\otimes \\R^3$ has dimension $6$.\n",
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