Page 50 - A Course in Linear Algebra with Applications

34 Chapter Two: Systems of Linear Equations

The general theory of linear systems
Consider a set of m linear equations in n unknowns xi,x 2,
..., x n:

a lxxi
{ a 2\Xi + a i2x 2 + ••• • • + + a lnx n = = b\ b 2
+
+
•
a 22x 2
a 2nx n
&mixi + a m2x 2 + • • • + a mnx n = b m
By a solution of the linear system we shall mean an n-column
vector
Xl
f \

x 2
\x J
n
such that the scalars x\, x 2, ..., x n satisfy all the equations
of the system. The set of all solutions is called the general
solution of the linear system; this is normally given in the form
of a single column vector containing a number of arbitrary
quantities. A linear system with no solutions is said to be
inconsistent.
Two linear systems which have the same sets of solutions
are termed equivalent. Now in the examples discussed above
three types of operation were applied to the linear systems:
(a) interchange of two equations;
(b) addition of a multiple of one equation to another
equation;
(c) multiplication of one equation by a non-zero scalar.
Notice that each of these operations is invertible. The critical
property of such operations is that, when they are applied
to a linear system, the resulting system is equivalent to the
original one. This fact was exploited in the three examples
above. Indeed, by the very nature of these operations, any