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.
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: $$\begin{align} \alpha \otimes v &amp;= (e^3 - 4e^1)\otimes(-2e_1+e_3)\\ &amp;= -2 e^3\otimes e_1 + e^3\otimes e_3 + 8e^1\otimes e_1 - 4 e^1\otimes e_3.\end{align}$$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: $$(\alpha \otimes v)(w,f) = -2w^3f_1 + w^3f_3 + 8f^1w_1 - 4f^1w_3.$$
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 ?