Vector space/Tensor product/Dimension/Fact

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ denote finite-dimensional vector spaces over ${}K$.

Then the dimension of the tensor product is

${}\dim _{K}{\left(V_{1}\otimes _{}\cdots \otimes _{}V_{n}\right)}=\dim _{K}{\left(V_{1}\right)}\cdots \dim _{K}{\left(V_{n}\right)}\,.$

Proof, Write another proof