space L. As a partial space of the vector space, formed out of all (n,1)-matrices with elements out of K, L has a dimension s
n. Hence there exists a base x(1), ···, x(s)
of L,
and every vector of L, that is, every solution of Ax
= 0 has the form26. Homogeneous and non-homogeneous equations
The most general system of linear equations has the form
a11x1
+ ··· +a1nxn
= c1,
···············································
(26.1)
an1x1 + ··· +apnxn
= cp
which we will again represent by Ax = c. Let the coefficients aik and the right hand terms ci belong to a given body of numbers K and restrict the solutions xi to K. You call (26.1) a system of homogeneous equations, when all the c1, ···, cp = 0, a non-homogeneous system when the right hand side of at least one equation differs from 0. You obtain from a non-homogeneous system the associated homogeneous system by setting all right hand sides equal to zero.
A homogeneous system Ax = 0 has always the trivial solution x = 0. As has been seen in Section 12, it certainly can happen that this is the only solution. We will now assume that also the system Ax = 0 has non-trivial solutions. Beside every solution x1, also every multiple lx1 with an arbitrary number l is a solution, since in addition to Ax1 = 0 also A(lx1) = l·Ax1=0. Moreover, if x1 and x2 are are two solutions of the homogeneous system Ax = 0, then also their sum is a solution, since it follows from Ax1 = 0 and Ax2 = 0 that
A(x1 +x2) = Ax1 + Ax2 = 0.
These two facts
yield the general fact: If x1,
···, xk are solutions of
the homogeneous system Ax = 0,
then every linear combination l1x1+···+lkxk
is also a solution, whence: All solutions of the
system Ax = 0 form a
vector
space L.
As a partial space of the vector space, formed out of all (n,1)-matrices
with elements out of K, L has a
dimension s n. Hence there exists a base x(1), ···, x(s)
of L,
and every vector of L, that is, every solution of Ax
= 0 has the form
l1x(1)+···+lsx(s).
The set of all vectors of L is called the general solution of the homogeneous system of equations Ax = 0.
Next, consider a non-homogeneous system Ax = c. If y1 and y2 are two solutions of this system, that is, Ay1 = c and Ay2 = c, their difference yields
A(y1 - y2) = Ay1 - Ay2 = c - c = 0,
whence: The difference y1 - y2 is a solution of the homogeneous system Ax=0.
The difference of two solutions of a non-homogeneous system is a solution of the associated homogeneous system.
On the other hand, if Ay = c and Ax = 0, then
A(y + x) = Ay + Ax = c + 0 = c,
whence:
If you add to a solution of a non-homogeneous system an arbitrary solution of the associated homogeneous system, you obtain again a solution of the non-homogeneous system.
The last two statements yield:
You obtain all solutions of a non-homogeneous system by addition to a special solution of this system the general solution of the associated homogenous system.
We have thus reduced the theory of linear equations to the treatment of the two partial tasks:
1.
Determination of the general solution of the homogeneous system.
2. Determination of a specific solution of the
non-homogeneous system.
In particular, the second task involves the examination of the question whether a given non-homogeneous system has a solution.
The next two sections deal with these two partial tasks.