In the wiki on bilinear forms, the universal product says that to each bilnear map $b : V\times V \rightarrow \mathbb{R}$ we can associate a linear map $\hat{b} : V\otimes V \rightarrow \mathbb{R}$, namely $\hat{b}(v\otimes w)=b(v,w)$. But they claim that this association is not canonical. I don't understand why, since this seems like the obvious choice for the correspondence, no?

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    $\begingroup$ The WP is wrong here, at least on the "canonical" part. As for "unique": sure, you can twist it by automorphisms, but the definition isn't to blame for that. $\endgroup$ – darij grinberg Apr 15 at 17:24
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    $\begingroup$ I fixed the Wikipedia article. $\endgroup$ – Oscar Cunningham Apr 15 at 20:50

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