# Tensor Product and its details

I'm having a tough time trying to understand tensor product and how to compute it. I don't know what to do when I have $\alpha\in V^*$, $\vec v\in V$ and I want to compute $\alpha\otimes\vec v$ or $\vec v\otimes\alpha$... For example, if I have $\alpha = (e^3-4e^1)$ and $\vec v = (-2\vec e_1+\vec e_3)$, $\alpha\otimes\vec v$ and $\vec v\otimes\alpha$ look like what?

The definition that my professor has taught me is this: given $\alpha,\beta\in V^*$ and $\vec u,\vec v\in V$ we have $(\alpha\otimes\beta)(\vec u,\vec v)\doteq \alpha(\vec u)\beta(\vec v)$. Thank you very much, folks.

• There are several definitions of the "tensor" product that will make $\alpha \times v$ look (at least superficially) very different. It would help us if you could write the definition that you're using, or at least point the textbook that you're using. Apr 23, 2017 at 22:33
• I meant to write $\alpha \otimes v$, of course. Please remember that you can edit your question to add details. Apr 23, 2017 at 22:40
• Thank you! I've forgotten to edit my question!! Apr 23, 2017 at 22:46
• Why has somebody downvoted this question (I have upvoted it) ? Caue Evangelista is a newcomer, that has tried to her best to explain her problem. Apr 23, 2017 at 22:47
• Thank very much for your kind words JeanMarie, I really don't understand this type of thing as well... Apr 23, 2017 at 22:51

## 2 Answers

The definition you wrote is for the tensor product of two linear functionals. You can consider the tensor product of a linear functional with a vector by "plugging in what makes sense", that is: $$(\alpha \otimes v)(w, \beta) \doteq \alpha(w)\beta(v),$$and so on. What you can do for now is write that as combination of "basic" tensors, using bilinearity of $\otimes$, as in: \begin{align} \alpha \otimes v &= (e^3 - 4e^1)\otimes(-2e_1+e_3)\\ &= -2 e^3\otimes e_1 + e^3\otimes e_3 + 8e^1\otimes e_1 - 4 e^1\otimes e_3.\end{align}If $f = f_i e^i$ and $w = w^j e_j$, you can see how $\alpha \otimes v$ acts explicitly by the above formula: $$(\alpha \otimes v)(w,f) = -2w^3f_1 + w^3f_3 + 8f^1w_1 - 4f^1w_3.$$

• Thank you soo much Ivo! I thought earlier that maybe $\alpha$$\otimes$$\vec v$ is done this way but I was not sure! Seriously, thank you so much! My best wishes! Apr 23, 2017 at 22:56
• By the way, you don't need to use all these dollars \\$ when typing mathjax here. See my edit to your question. Apr 23, 2017 at 22:57
• Oh, it's much more easy to read in your way! Thanks! Apr 23, 2017 at 23:00

Probably you are using for aeV* and veV that aoxv is the bilinear form on VxV* where v acts on b* in V* by point evaluation - v (b*) = b*(v) (thus identifying V** with V so aoxv(w,b*) = a(w)b*(v) .This works as long as V is finite dimensional .Hope that helps .What book are you using ?

• I'm using the Arfken's book "Mathematical Methods for Physicists" Apr 23, 2017 at 23:07