Consider the tensor product $V\bigotimes W$, and $v_1\otimes w_1$, $v_2\otimes w_2$ in $V\bigotimes W$, where $V$ and $W$ are $F$-algebras.
Can we add or multiply the pure tensors as follows?
$(v_1\otimes w_1)(v_2\otimes w_2)=v_1v_2\otimes w_1w_2$
$(v_1\otimes w_1)+(v_2\otimes w_2)=(v_1+v_2)\otimes (w_1+w_2)$
Something tells me that the above is not quite right?
Thanks for help!
Best Answer
The multiplication is often defined that way for associative algebras by extending linearly, and it works, but the addition is not correct. In general not every element of $V\otimes W$ can be written as $v\otimes w$ for $v\in V$ and $w\in W$. Instead we have the rules $$(v_1+v_2)\otimes w=v_1\otimes w+v_2\otimes w$$ and similarly for $v\otimes (w_1+w_2)$. It may very well be in a given case that $v_1\otimes w_1+v_2\otimes w_2$ can get no simpler than that, and of course more than two terms may be required in general.