----- > [!proposition] Proposition. ([[vector space basis iff minimal spanning set]]) > A [[basis]] of a [[free module]] $M$ over *any* [[ring]] $R$ is necessarily [[maximal|minimally]] [[submodule generated by a subset|generating]]. If $R=k$ is a [[field]] (so that $M=V$ is a [[vector space]]), then the converse holds: $B$ is a [[basis]] of $V$ if and only if $B$ is a minimally [[submodule generated by a subset|generating]] subset of $V$. ^proposition > [!specialization] Statement for finite-dimensional vector spaces. > [[Definition Template]] > Suppose $V$ is [[vector space#Finite-Dimensional Vector Space|a finite-dimensional vector space]]. A list $B \subset V$ it a [[basis]] of $V$ if and only if it is a minimal [[spans|spanning set]]**. > > (Here "minimal [[spans|spanning set]]" means that removing any $v \in V$ from $B$ will cause $B$ to fail to [[submodule generated by a subset]] $V$.) > [!proof] Proof for finite-dimensional vector spaces. > Suppose $\dim V=n$. Let $B \subset V$. > > Suppose $B$ be is **minimal [[spans|spanning set]] subset** of $V$. By [[spanning list of the right length is a basis]] we're done with one direction if we can show $\len \ B = n$. $B$ [[every generating set contains a basis|can be reduced to a basis]] of $V$ by removing from $B$ some number $k \in \nn$ of [[vector]]s. Since $B$ is minimally [[spans|spanning]], $k=0$(otherwise the resultant list would not be a [[basis]]). So $\len \ B=n$ as required. > > Conversely suppose $B$ is a [[basis]] of $V$. Remove from $B$ a [[vector]] $v \in V$, note that $B \setminus v$ is [[linearly independent]] because $B$ is. By [[length of linearly independent list is at most length of spanning list]], since $B$ is [[linearly independent]] with length $n$, $B \setminus v$ with length $n-1$ must not [[submodule generated by a subset]] $V$. Hence $B$ is a minimal [[spans|spanning set]]. > [!proof]- Proof. ([[vector space basis iff minimal spanning set]]) > Bring over. ----- #### ---- #### 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 > ```