Where can I find a proof of the following general Steinitz exchange lemma:
Let $B$ be a basis of a vector space $V$, and $L\subset V$ be linearly independent. Then there is an injection $j:L\rightarrow B$ such that $L\cup(B\setminus j(L))$ is a disjoint union and a basis of the vector space $V$.
Or can someone suggest a proof of this result? Probably using Zorn's lemma.
Thank you!
Best Answer
Let $W$ denote the subspace generated by $L$, $\mathcal B$ denote the set of subsets of $B$ whose images in $V/W$ are linearly independent. There is an obvious partial order on $\mathcal B$, namely the one induced by inclusion. Apply Zorn's Lemma to obtain a maximal element of $\mathcal B$, denoted by $C$, then the disjoint union $C\cup L$ forms a basis of $V$, just as $B=C\cup(B\setminus C)$ do. So there must be a bijection $j:L\xrightarrow{\sim}B\setminus C$. This $j$ will fulfill the requirement.