Vector space/Tensor product/Generating system/Basis/Fact

Let $K$ be a field, and let $V_{1}, \dots, V_{n}$ denote vector spaces over $K$ . Let $J_{1}, \dots, J_{n}$ be index sets, and let

v_{i j}, j \in J_{i},

be vectors in $V_{i}$ . Then the following statements hold.

If each family is a generating system of $V_{i}$ , then the family
$v_{1 j_{1}} \otimes_{} \dots \otimes_{} v_{n j_{n}} with (j_{1}, \dots, j_{n}) \in J_{1} \times \dots \times J_{n}$

is a generating system of $V_{1} \otimes_{} \dots \otimes_{} V_{n}$ .
If each family is linearly independent in $V_{i}$ , then the family
$v_{1 j_{1}} \otimes_{} \dots \otimes_{} v_{n j_{n}} with (j_{1}, \dots, j_{n}) \in J_{1} \times \dots \times J_{n}$

is linearly independent in $V_{1} \otimes_{} \dots \otimes_{} V_{n}$ .
If each family is a basis of $V_{i}$ , then the family
$v_{1 j_{1}} \otimes_{} \dots \otimes_{} v_{n j_{n}} with (j_{1}, \dots, j_{n}) \in J_{1} \times \dots \times J_{n}$

is a basis of $V_{1} \otimes_{} \dots \otimes_{} V_{n}$ .