Basis for the tensor product/Statement

Statement

Let $\mathbb{F}$ be a field and let $\big((V_i,\mathbb{F})\big)_{i\eq 1}^k$ be a family of finite dimensional vector spaces. Let $n_i:\eq\text{Dim}(V_i)$ and $e^{(i)}_1,\ldots,e^{(i)}_{n_i}$ denote a basis for $V_i$, then we claim:

• $$\mathcal{B}:\eq\left\{e^{(1)}_{i_1}\otimes\cdots\otimes e^{(k)}_{i_k}\ \big\vert\ \forall j\in\{1,\ldots,k\}\subset\mathbb{N}[1\le i_j\le n_j]\right\}$$

Is a basis for the tensor product of the family of vector spaces, $V_1\otimes\cdots\otimes V_k$

Note that the number of elements of $\mathcal{B}$, denoted $\vert\mathcal{B}\vert$, is $\prod_{i\eq 1}^kn_i$ or $\prod_{i\eq 1}^k\text{Dim}(V_i)$, thus:

• $\text{Dim}(V_1\otimes\cdots\otimes V_k)\eq\prod_{i\eq 1}^k n_i$