Basis of a tensor product

Statement

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$. Let $(e_i)_{i \in I}$ be a basis of $V$ and let $(f_j)_{j \in J}$ be a basis of $W$. Then the family $(e_i \otimes f_j)_{(i, j) \in I \times J}$ is a basis of the tensor product $V \otimes W$.

Remarks

So each vector of $V \otimes W$ is uniquely $\sum_{i,j} c_{ij}\, e_i \otimes f_j$ with only finitely many nonzero $c_{ij} \in F$. For $v = \sum_i a_i e_i$ and $w = \sum_j b_j f_j$, $v \otimes w = \sum_{i,j} a_i b_j\, e_i \otimes f_j$. Thus the coefficients of a vector $v \otimes w$ form a matrix of rank at most $1$.

Explanations

L1 human

From a basis of each factor, we get a basis of the tensor product. Its vectors are all the products of a basis vector of $V$ and a basis vector of $W$. A vector of $V \otimes W$ is like a matrix of coefficients.