Vector space/Change of base field/Generating system/Basis/Fact

Let ${}K$ be a field, ${}V$ be a ${}K$-vector space, and ${}K\subseteq L$ be a field extension. Let ${}v_{i}$, ${}i\in I$, be a family of vectors in ${}V$. Then the following statements hold.

1. The family ${}v_{i}$, ${}i\in I$, is a ${}K$-generating system of ${}V$ if and only if ${}1\otimes v_{i}$, ${}i\in I$, is an ${}L$-generating system of ${}L\otimes V$.
2. The family ${}v_{i}$, ${}i\in I$, is ${}K$-linearly independent in ${}V$ if and only if ${}1\otimes v_{i}$, ${}i\in I$, is linearly independent (over ${}L$) in ${}L\otimes V$.
3. Die family ${}v_{i}$, ${}i\in I$, is a ${}K$-basis of ${}V$ if and only if ${}1\otimes v_{i}$, ${}i\in I$, is an ${}L$-basis of ${}L\otimes V$.

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