---- > [!definition] Definition. ([[equivalent matrices]]) > Let $R$ be an [[integral domain]]. Two [[matrix|matrices]] $P,Q \in \mathcal{M}_{m,n}(R)$ are said to be **equivalent** if they represent the same [[linear map|homomorphism]] of [[free module|free]] [[module|modules]] $R^{\oplus n} \to R^{\oplus m}$ up to a choice of [[basis]]. > This is manifestly an [[equivalence relation]]. (Here, $\mathcal{M}_{m,n}(R)$ denotes the [[vector space of m-by-n matrices|R-module of m-by-n matrices]].) ^definition > [!equivalence] > - $P$ and $Q$ are equivalent if and only if there exist [[inverse matrix|invertible]] $M$ and $N$ such that $Q=MPN$: this is exactly what [[change of basis formula]] tells us. > - If $R=k$ is a [[field]] then we are in great shape: since [[elementary operations produce equivalent matrices]] and [[the general linear group over a field is generated by elementary matrices]], over a [[field]] $P$ and $Q$ will be equivalent if and only if each can be obtained from the other via a sequence of [[elementary operation|elementary operations]]. > - Taking this a bit further in [[Gaussian elimination]], we see that matrices are equivalent if and only if they have the same [[rank]]. ^equivalence > [!NOTE] Remark. > In light of this definition, one would say that — properly speaking — a [[linear map|homomorphism]] $\alpha:F \to G$ of [[free module|free]] $R$-[[module|modules]] is not represented by one [[matrix]] as much as by a whole *[[equivalence class]]* of [[matrix|matrices]]. ---- #### ---- #### 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 > ```