# Tensor Algebra help

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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Maybe you could say what $S$ is supposed to be. – a.r. Feb 13 '11 at 7:46
sorry I should have elaborated further. S is suppose to be a tensor map while a and b are supposed to be vectors. – user7019 Feb 13 '11 at 7:53
What do you mean by "tensor map"? Is it linear? What is its domain and range? – Jim Belk Feb 13 '11 at 8:23
upon further looking at the book, it looks like a linear map. But I guess what they were just trying to get across is that it is a Tensor. such that v = Su. The hint in the back of the book says apply each side of the identity to an arbitrary vector v. – user7019 Feb 13 '11 at 8:39
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. – wildildildlife Feb 13 '11 at 12:10

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 $S(a \otimes b) = (Sa) \otimes b$.
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, $a$.