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. The 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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