General Steinitz exchange lemma

Question

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!

Answer 1

Choose (by using Zorn's Lemma) a subset $B'$ of $B$ maximal with the property that $\langle L\rangle\cap \langle B'\rangle=0$. Then $\langle L\rangle+\langle B'\rangle=V$: if $b\in B-B'$ then $\langle L\rangle\cap \langle B'\cup\{b\}\rangle\ne0$ hence there are $l_i\in L$, and $b_i'\in B'$ such that $\sum\alpha_il_i=\sum_i\beta_ib_i'+\beta b$ with $\beta\ne0$, so $b\in \langle L\rangle+\langle B'\rangle$.

Now use that any two bases of a vector space are equipotent. Let $V'=\langle B'\rangle$. The quotient vector space $V/V'$ has the following bases: $L$ and $B\setminus B'$. Then there is a bijection $f:L\to B\setminus B'$ which composed by the inclusion $i:B\setminus B'\to B$ gives us the desired injection, that is, $j=i\circ f$.

Answer 2

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.