This is essentially a follow-up on this question here. Specifically, I'm studying the construction of the tensor product of two vector spaces as indicated by the OP's 2nd definition. Namely, that the tensor product of two vector spaces $V$ and $W$ is the quotient space $V \times W / E$ where $E$ is the subspace of $V \times W$ generated by linear combinations of the form
$ (v_1 + v_2, w) - (v_1, w) - (v_2, w) $ $ (v, w_1 + w_2) - (v, w_1) - (v, w_2) $ $ (av, w) - a(v,w) $ $ (v,aw) - a(v,w) $
We also denote an element of this quotient space by $v \otimes w$. Now, I'm trying to understand precisely why it follows from this construction that, for example, $(v_1 + v_2) \otimes w = v_1 \otimes w + v_2 \otimes w$. I believe it follows from the following argument:
First, we note that
$ (v_1 + v_2, w) - (v_1, w) - (v_2, w) = (v_1 + v_2, w) - ((v_1, w) + (v_2, w)) $
By definition of what it means to be a quotient space, this last equation means that
$ [ (v_1 + v_2, w)] = [(v_1, w) + (v_2, w)] $
where $[ \cdot ]$ denotes the induced equivalence class. But, according to the definition of the $\otimes$ operator, we can express this equality as
$ (v_1 + v_2) \otimes w = v_1 \otimes w + v_2 \otimes w $
Is this argument valid and is this the appropriate way to think about these things?