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I am working on my homework for a class, and I am really stuck

the question I was given was:

prove that S(a ⊗ b) = (Sa) ⊗ b

Does anyone have any tips on how to solve this?

For further explanation this is problem 6a from this book on the page the link goes to

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    The terminology of the book does not seem to be standard. In order to get help, you really need to elaborate more on what S, a, and b precisely are. And v and u.2011-02-13

2 Answers 2

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The important identity that is missing from the Google Books preview is $(a \otimes b) v = (b \cdot v)a$. Using this, and looking at the image of a vector $v$ under $S(a \otimes b)$ you get $\begin{aligned} S(a \otimes b) v &= S((b \cdot v)a)\\ &= (b \cdot v) (Sa)\\ &= ((Sa) \otimes b) v \end{aligned}$ Hence $S(a \otimes b) = (Sa) \otimes b$.

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I think this is an easy problem using the universal property of tensor products provided that I can understand the question, it seems that the precise definition in the linked book is blocked and it is truly weird o find S operating on both tensors , $a$ tensor product $b$ and vectors, $a$.
As far as I am concerned, it is either a typing problem or an unexplained notation arising error.