Tensor-product basis

If $\{e_i\}$ and $\{f_j\}$ are bases of finite-dimensional vector spaces $V,W$, then $\{e_i\otimes f_j\}$ is a basis of $V\otimes W$. Bilinearity gives spanning, and applying products of dual coordinate functionals gives independence. Thus $\dim (V\otimes W)=\dim V\dim W$.