---- > [!theorem] Theorem. ([[the general linear group over a field is generated by elementary matrices]]) > Let $R=k$ be a [[field]] and let $n \geq 0$ be an integer. Then the [[general linear group]] $\text{GL}_{n}(k)$ is [[generating set of a group|generated by]] [[elementary operation|elementary matrices]]. > ^theorem > [!proof]- Proof. ([[the general linear group over a field is generated by elementary matrices]]) > Let $A \in \text{GL}_{n}(k)$ and consider the algorithm discussed in [[Gaussian elimination]]. Observe that at each stage, the [[matrix]] $A'$ is [[inverse matrix|invertible]], e.g. in light of column [[linearly independent|linear independence]] considerations (since $A$ is [[inverse matrix|invertible]]). This means that we can always assume some entry of the first column is nonzero (by [[permutation matrix|performing row switch]] if necessary, an [[elementary operation]]) can assume $a_{11} \neq 0$ in any (sub)matrix. In light of this, [[Gaussian elimination]] (on rows + columns, so maybe not GE in the standard sense) reduces $A$ to $I_{n}$ via a sequence of [[elementary operation|elementary matrices]] multiplied on the right/left: $I_{n}=M AN$ for $M,A,N \in \text{GL}_{n}(k)$, where $M,N$ are products of [[elementary operation|elementary matrices]]. But then $A=M ^{-1} N ^{-1}$ is itself a product of [[elementary operation|elementary matrices]], yielding the statement. ---- #### ----- #### 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 > ```