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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 $V$ and $W$ are [[isomorphic|0B8T10CE7N]] if and only if $\\dim V = \\dim W$, where $\\dim$ is the [[dimension|03GSZ8RTZJ]].\n\n## Remarks\n\nAn isomorphism sends a basis to a basis, so it keeps the dimension.\nConversely, a bijection between bases of $V$ and $W$ extends to a linear isomorphism.\nThe statement holds for all vector spaces, with dimension as a cardinal number.\nFor example, $M_{2 \\times 3}(F)$, $F^6$ and $F[x]_{\u003c6}$ are pairwise isomorphic.\n\n## Explanations\n\n### L1 human\n\nUp to isomorphism, a finite-dimensional vector space is determined by one number: its dimension.\nAll three-dimensional real spaces behave like $\\R^3$.\nOnly the names of the vectors differ.\n",
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