Linear system/Simultaneously/Remark

Sometimes, we want to solve a simultaneous system of linear equations of the form

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&=&c_{1}&(=&d_{1},&=&e_{1},\ldots )\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&=&c_{2}&(=&d_{2},&=&e_{2},\ldots )\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&=&c_{m}&(=&d_{m},&=&e_{m},\ldots )\,.\end{matrix}}$

The goal is to find the solutions of the corresponding inhomogeneous linear systems for different vectors. In principle, we could consider independent linear systems, and solve them. However, it is smarter to perform those manipulations that we do on the left-hand side to achieve upper triangular form, simultaneously with all the vectors on the right-hand side. An important special case, for ${}n=m$, is when the vectors are the standard vectors, see fact.