Linear system/Matrix/Manipulations/Remark

A linear system can be written briefly as

${}Ax=c\,$

with an ${}m\times n$-matrix ${}A$ and an ${}m$-tuple ${}c$. The manipulations at the equations that we do in the elimination procedure, can be performed directly for the matrix, or for the extended matrix that arises when we extend ${}A$ by the column ${}c$. Essentially, we replace a row by the sum of the row with a multiple of another row. This has the advantage that we do not have to write down the variables. However, one should then not swap the variables. At the end, the arising matrix in echelon form can be interpreted again as a linear system.