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General Research Into the System of Linear Algebraic Equations

Lecture 3. General Research Into the System

of Linear Algebraic Equations

3.1. General information about systems of linear equations

In Lecture 1 and Lecture 2 we have studied methods of solving systems of linear equations.
But these methods can be used only for systems that have the same number of unknowns and
equations (three in most cases). In this section we will study the general method of solving
systems of linear equation, in which the number of unknowns and equations doesn’t coincide.
Let’s consider the system of m linear equations with n unknowns:

 a 11 x 1 + a 12 x 2 + ... + a 1n x n = b 1 ,

a 21 x 1 + a 22 x 2 + ... + a 2n x n = b 2 ,

(3.1)
 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .,


a m1 x 1 + a m2 x 2 + ... + a mn x n = b m .
A set of unknowns (x , x , ..., x ) is called a solution of system (3.1) if on substituting these
′
′
′
m
2
1
unknowns into (3.1) we will get equalities. When the system has at least one solution it is called
compatible and noncompatible when it has no solutions. The system, which has only one solution,
is called determined; the system, which has more than one solution, is called undetermined. Two
compatible systems are called equivalented if each solution of the first system is a solution for
the second one and vice versa.

3.2. Theorem on the existence of a system of linear equations solution

Let’s consider the system (3.1).

Theorem 3.1 (Theorem of Kronecker & Capelli).
In order for system (3.1) to be compatible it is necessary and sufficient that the rank of the main

matrix
 
a 11 a 12 . . . a 1n
 a 21 a 22 . . . a 2n 
A =  
 . . . . . . . . . . . . 
a m1 a m2 . . . a mn
equals to the rank of extended one, obtained from the main one adding a column of free members:

 
a 11 a 12 . . . a 1n b 1
a a . . . a b
 
A =  21 22 2n 2  .
 . . . . . . . . . . . . . . . 
a m1 a m2 . . . a mn b m

PROOF. Necessity. Let’s assume that the system (3.1) is compatible and (x , x , . . . , x ) is
′
′
′
m
2
1
one of its solutions. We have to prove that r(A) = r(A). Let’s substitute this solution into the system
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