Can’t say much about transformers, but the tensor product definition seems off. There can be many elements in V⊗W that aren’t expressible as v⊗w, only as a linear combination of multiple such. That can be seen from dimensionality: if v and w have dimensions n and m, then all possible pairs can only span n+m dimensions (Cartesian product), but the full tensor product has nm dimensions.
Here’s an explanation of tensor products that I came up with sometime ago in an attempt to make it “click”. Imagine you have a linear function that takes in two vectors and spits out a number. But wait, there are two natural but incompatible ways to imagine it:
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f(a,b) + f(c,d) = f(a+c,b+d), linear in both arguments combined. The space of such functions has dimension n+m, and corresponds to Cartesian product.
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f(a,b) + f(a,c) = f(a,b+c) and also f(a,c) + f(b,c) = f(a+b,c), in other words, linear in each argument separately. The space of such functions has dimension nm, and corresponds to tensor product.
It’s especially simple to work through the case n=m=1. In that case all functions satisfying (1) have the form f(x,y)=ax+by, so their space is 2-dimensional, while all functions satisfying (2) have the form f(x,y)=axy, so their space is 1-dimensional. Admittedly this case is a bit funny because nm<n+m, but you can see how in higher dimensions the space of functions of type (2) becomes much bigger, because it will have terms for x1y1, x1y2, etc.
I think the right procedure works something like this: 1) Tenants notice that one of them has trashed the garden, and tell the landlord who. 2) The landlord tells the offending tenant to clean up or they’ll be billed. 3) If the offending tenant doesn’t clean up, the cleaning fee gets added to their next rent bill.
In your case it seems like the offending tenant wasn’t pointed out. Maybe because other tenants didn’t care, or maybe some tenants had a mafia mentality and made “snitching” unsafe. Either way, you were right to move away.