(related to Chapter: Vectors Revised - Vector Spaces)
Note that the elementary Gaussian operations $I$ and $II$ correspond to the scalar multiplication and the vector addition, followed by a scalar multiplication. Moreover, if we consider a system of linear equations (SLE) in a vector space $V$, then the following can be said about its solutions:
If the vector $u$ is a solution inhomogeneous system of linear equations $Ax=b,$ then all of its solutions are given by the vectors $y:=u\pm h,$ where $h$ is the solution of the homogeneous system of linear equations $Ah=0.$
In fact, the above statement reformulates in a much shorter notation a theorem stating the relationship between the solutions of homogeneous and inhomogeneous SLEs, which was proved above.