----- > [!proposition] Proposition. ([[every spanning set reduces to maximal linearly independent set]]) > Let $R$ be an [[integral domain]], and let $M$ be a [[free module|free]] $R$-[[module]]; assume that $M$ is [[submodule generated by a subset|generated]] by $S$: $M=\langle S \rangle$. Then $S$ contains a [[maximal]] [[linearly independent]] subset of $M$. ^proposition > [!specialization] For finite-dimensional vector spaces. > If $V$ is a finite-dimensional [[vector space]] over [[field]] $k$, this result specializes to [[length of linearly independent list is at most length of spanning list]]. ----- #### ---- #### 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 > ```