Unclear Construction of Basis for Tensor Product

Question

My problem lies in page 363 of Steven Roman's Advanced Linear Algebra (Here's a link). The author says that for each ordered pair $(e_i,f_j)$ where $\left\{e_i\right\}_{i\in I},\left\{f_j\right\}_{j\in J}$ are respective bases for vector spaces $U,V$, we "invent a new formal symbol" $e_i\otimes f_j$. He then defines $T$ (that will soon be taken as $U\otimes V$) to be the vector space with basis $$\mathcal{D}=\left\{ e_i\otimes f_j \mid i\in I,j\in J\right\}$$ Without further remarks. Why is $\mathcal{D}$ a basis? We don't know anything at all about $e_i\otimes f_j$ as they're just symbols..

Thanks in advance!

Answer 1

He takes $D$ to be that set $D = \{e_i \otimes f_j\}$, where all elements are just formal symbols.

Then he is defining $U \otimes V$ to be the free vector space $\text{Free}(D)$, which is the set of all functions $f : D \to \mathbb{F}$ which are zero except at finitely many values of $D$. If you read about free vector spaces, you'll see that $D$ is indeed a basis of $\text{Free}(D)$ (basically by definition).

edit: Here are some references which I recommend on this subject: Dummit and Foote's Abstract Algebra and Werner Greub's Multilinear algebra.