Linear system/Solution method/Several remarks/Section

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.

We discuss briefly some further methods to solve a linear system.

Remark

Another method to solve a linear system is the substitution method. Here, the variables are also successively eliminated, but in another way. If we want to eliminate the variable ${}x_{1}$, then we look at an equation, say ${}G_{1}$, where ${}x_{1}$ occurs with a coefficient different from ${}0$. In this equation, we isolate ${}x_{1}$ and get a new equation of the form

$G_{1}':\,\,\,x_{1}=F_{1},$

where ${}x_{1}$ does not occur in ${}F_{1}$. Then in all other equations ${}G_{2},\ldots ,G_{m}$, we replace the variable ${}x_{1}$ by ${}F_{1}$, and obtain (after some simplifications) a linear system ${}G_{2}',\ldots ,G_{m}'$ without the variable ${}x_{1}$, which is, together with ${}G_{1}'$, equivalent with the original system.

Remark

Another method to solve a linear system is the equating method. Here, the variables are also successively eliminated, but in another way. In this method, in every equation ${}G_{i}$, ${}i=1,\ldots ,m$, we isolate one fixed variable, say ${}x_{1}$. Suppose that (after reordering) ${}G_{1},\ldots ,G_{k}$ are the equations where the variable ${}x_{1}$ occurs with a coefficient different from ${}0$. These equations are brought into the form

$G_{i}':\,\,\,x_{1}=F_{i},$

where in ${}F_{i}$, the variable ${}x_{1}$ does not occur. The linear system consisting in

$G_{1}',F_{1}=F_{2},F_{1}=F_{3},\ldots ,F_{1}=F_{k},G_{k+1},\ldots ,G_{m}$

is equivalent to the original system. We continue with this system without ${}G_{1}'$.

Remark

The methods described in fact, remark, and remark to solve a linear system differ with respect to speed, strategic conception, complexity of the coefficients, error-proneness. In the elimination method, the systematic reduction of the number of variables (reduction of dimension) is obvious, and it is unlikely to make mistakes (except for miscalculations). It is always clear how to continue. However, these advantages emerge starting with three variables. For two variables, it does not make a difference what method we choose.

The evaluation of the methods depends also on the features of the concrete system. Such features should be taken into account in order to find "short-cuts“ to the solution. The adequate choice of the solution method appropriate for the given problem is called adaptivity (a concept which is used in the didactic context with different meanings). If, for example, one row of the system has the form ${}x=3$, then one should recognize that a part of the solution can be read off immediately, and one should not add to this row any other row. Here, one should replace in the other rows everywhere ${}x$ by ${}3$, and continue then. Or: if four equations are given, where in two equations only the variables ${}x$ and ${}y$ appear, and where in the two other equations only the variables ${}z$ and ${}w$ appear, then one should realize that, in principle, two unrelated linear systems are given, each in two variables, and these should be solved independently. Or: it might be that a small subsystem of the system guarantees that there is no solution at all. Then only this has to be worked out, there is no need to consider the other equations. And: consider the exact question! If the question is whether a certain tuple is a solution, then we only have to plug this tuple into the equations, no manipulations are necessary.

Remark

A system of linear inequalities over the rational numbers or over the real numbers is a system of the form

${\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}&\star &c_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}&\star &c_{2}\\\vdots &\vdots &\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}&\star &c_{m}\,,\end{matrix}}$

where ${}\star$ might be ${}\leq$ or ${}\geq$. It is considerably more difficult to find the solution set of such a system than in the case of equations. In general, it is not possible to eliminate the variables.