Solving Systems of Linear Equations

Solving systems of linear equations

7.1 Introduction
Let a system of linear equations of the following form:

a11 x1 a21 x1





a12 x2 a22 x2


ai1x1  ai 2 x2

 am1 x1  am2 x2





a1n xn a2 n x n





   ain xn


   amn xn



b1 b2 

bi

 bm

(7.1)

be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m; k = 1, 2, ..., n) are the coefficients, bi (i = 1, 2, ..., m) are the free terms of the system. In matrix notation, this system has the form:
Ax  b ,

(7.2)

where A is the matrix of coefficients of the system (the main matrix), A = [aik]mn, b is the column vector of the free terms, bT  [b1 , b2 , ... , bm ] , x is the column vector of the unknowns, xT  [ x1 , x2 , ... , xn ] ; the symbol () T denotes transposition.

It is assumed that aik and bi are known numbers. An ordered set, {x1, x2, ..., xn}, of real numbers satisfying (7.1) is referred to as the solution of the system, and the individual numbers, x1, x2, ..., xn, are roots of the system.

A system of linear equations is:


consistent - if it has at least one solution. At the same time it can be
-



determined - if it has exactly one, unique solution, undetermined - if it has infinitely many solutions;

inconsistent - if it does not have any solution.

The further considerations will be limited to most frequently met in technical problems non-homogeneous systems of equations, i.e., systems that satisfy:

m

b

2 i  0.

(7.3)

i 1

Conditions for existing of solution of such systems give the following theorem:

Theorem 7.1 (Kronecker-Capelly)
The system of equations (7.1) is solvable if and only if, in notation (7.2):

rank(A) = rank(B) = r,

where B is the extended matrix created by inserting vector b as the n+1 column of matrix A. At the same time:


there exists exactly one, unique solution if r = n;



there are