Consider two vector spaces $E$ and $F$ over the same field $K$. Now to form the tensor product $E\otimes F$ we typically take a particular vector space $V_1$ and quotient it. The vector space $V_1$ is from page 265 here, is defined as $V_1=\oplus_{(e,f)\in E\times F}K(e,f)$. My questions are:
(1) Is $K(e,f)$ the set of all formal expressions of the form $\alpha\cdot(e,f)$? If so, how is this a vector space? Alternatively is $K(e,f)$ the span of $(e,f)$ in the vector space $E\times F$?
(2) How do we visualize the direct sum? The way I see it, $K(e,f)$ the span of $(e,f)$ and the direct sum consists of all those elements of $\prod K(e,f)$ for which all but finitely many coordinates are zero. But then how are the basis elements of $V_1$ all the elements of $E\times F$?
Thanks