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  "body": "\n## Summary\n\nThe symmetric algebra of a vector space $V$ is the [[tensor algebra|P4J5JZDW0B]] of $V$ made commutative.\nIt is the most general commutative algebra generated by $V$.\nIf $V$ has a finite basis $x_1, \\ldots, x_n$, it is the [[polynomial ring|0SDC934R52]] $F[x_1, \\ldots, x_n]$, so it gives a coordinate-free description of polynomials.\n\n## Explanations\n\n### L1 human\n\nIn the tensor algebra the order of the factors matters, so $x \\otimes y$ and $y \\otimes x$ are different.\nThe symmetric algebra forgets the order: products of vectors behave like products of numbers.\nWith a basis $x, y$ of $V$, the elements of degree $2$ are the combinations of $x^2$, $xy$ and $y^2$.\nSo the symmetric algebra is the ring of polynomials in the basis vectors.\n",
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