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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nThe space $V$ is *finite-dimensional* when it has a finite [[spanning set|088RYKAWK9]].\nOtherwise $V$ is *infinite-dimensional*.\n\n## Remarks\n\nA finite-dimensional space has a finite basis, because [[every spanning set contains a basis|06MRNP4BNT]].\nThe spaces $F^n$ and $M_{m \\times n}(F)$ are finite-dimensional.\nThe polynomial ring $F[x]$ is infinite-dimensional: finitely many polynomials have a bounded degree, so they do not span $F[x]$.\nEach [[subspace|00YC1D652K]] of a finite-dimensional space is finite-dimensional.\n\n## Explanations\n\n### L1 human\n\nA space is finite-dimensional when finitely many vectors are enough to build every vector.\nThe plane needs two vectors, and ordinary space needs three.\nThe space of all polynomials needs infinitely many, because the degrees have no bound.\n",
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