----- > [!proposition] Proposition. ([[existence maximal linearly independent subsets]]) > Let $R$ be a [[commutative ring]], $M$ an $R$-[[module]] and let $S \subset M$ be a [[linearly independent]] subset. Then there exists a maximal [[linearly independent]] subset of $M$ containing $S$. ^proposition > [!specialization] For finite-dimensional vector spaces. > If $V$ is a finite-dimensional [[vector space]] over [[field]] $k$, in light of [[vector space basis iff maximal linearly independent]] this result specializes to the result [[every linearly independent list extends to a basis]]. > [!NOTE] Remark. > As a sort of dual to this statement, [[every spanning set reduces to maximal linearly independent set]]. > [!proof]- Proof. ([[existence maximal linearly independent subsets]]) > Consider the family $\mathscr{S}$ of [[linearly independent]] subsets of $M$ containing $S$. Since $S$ is [[linearly independent]], $\mathscr{S} \neq \emptyset$. By [[Zorn's lemma]], it is enough to show that every [[poset|chain]] in $\mathscr{S}$ has an [[upper bound]] in $\mathscr{S}$. Well, the union of all sets comprising a given chain is again [[linearly independent]]: any [[linear combination]] of its elements has finitely many terms and is therefore a [[linear combination]] of elements in one of the sets comprising the chain (thus vanishes iff its coefficients do). ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```