  Let $R$ be a ring with unit. Passing to the colimit with respect to the
standard inclusions $GL(n,R) \to GL(n+1,R)$ (which add a unit vector as new
last row and column) yields, by definition, the stable linear group $GL(R)$;
the same result is obtained, up to isomorphism, when using the "opposite"
inclusions (which add a unit vector as new first row and column). In this note
it is shown that passing to the colimit along both these families of inclusions
simultaneously recovers the algebraic $K$-group $K_1(R) = GL(R)/E(R)$ of~$R$,
giving an elementary description that does not involve elementary matrices
explicitly.
