>>12489392Kind of yes. But its not e_i\otimes... which has these entries (because its the abstract vector) but the corresponding components a^{ijk} would look like that, if you choose e_i... as the basis.
Take for example R^2\otimesR^2
Let v,w be a basis of R2 then
v\otimes v, w\otimes w , v\otimes w and w\otimes v is the basis of the tensorproduct space.
Now IN COMPONENTS v and w look like (1,0) and (0,1)
So naturally we want to write as the gomponents of the productbasis (1,0\\0,0), (0,1\\0,0) etc. We see that our components have two indices but aside from that its exactly the same as in the untensored space. Now we cant just write this down, in fact you can proove that the components must look like this given the components in R2, its called dyadical product (or in german dyadisches produkt, idk if theres an english translation) so u see it is really simple. The fun begins when you start to tensor the dual space and look at the transformation behaviour of the components under basischsnges. Depending on the space they transform differently. This is then called co/contravariant trans. And physicists like to indicate this behaviour by placing the indices eitheir up or down. Its all way easier than its tought, i remember struggling but one day itll click trust me
