(All vector spaces are over a fixed field $F$).
Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space $V\otimes W$, along with a multilinear map $\pi:V\times W\to V\otimes W$ such that whenever there is multilinear map $A:V\times W\to X$ to any vector space $X$, there exists a unique linear map $\bar A:V\otimes W\to X$ such that $\bar A\circ \pi = A$.
Notation. We write $\pi(v,w)$ as $v\otimes w$.
First Question: Is the map $\pi :V\times W\to V\otimes W$ sujective?
I think it should be surjective since $\pi(V\times W)$ also satisfies the universal property of tensor product of $V$ and $W$. Since $V\otimes W$ is given by a universal property, it is unique upto a unique isomorphism and thus $\pi$ should be surjective.
Please correct me if I am wrong anywhere in the the above said things.
Main Question: Assuming all is well so far, I am confused in the following:
Let $(e_1,\ldots, e_m)$ and $(f_1,\ldots, f_n)$ be bases for $V$ and $W$ repectively.
I was wondering what member of $V\times W$ maps to $e_1\otimes f_1 + e_2\otimes f_2$.
Since $\pi$ is surjective, some member should map to it.
But I am lost as to how to find it.
Can somebody help?