Page 46 - A Course in Linear Algebra with Applications
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Chapter Two
SYSTEMS OF LINEAR EQUATIONS
In this chapter we address what has already been de-
scribed as one of the fundamental problems of linear algebra:
to determine if a system of linear equations - or linear system
- has a solution, and, if so, to find all its solutions. Almost
all the ensuing chapters depend, directly or indirectly, on the
results that are described here.
2.1 Gaussian Elimination
We begin by considering in detail three examples of linear
systems which will serve to show what kind of phenomena are
to be expected; they will also give some idea of the techniques
that are available for solving linear systems.
Example 2.1.1
xi - x 2 + x 3 + x 4 = 2
%i + X2 + x 3 - x 4 = 3
Xi + 3X2 + £3 — 3^4 = 1
To determine if the system has a solution, we apply
certain operations to the equations of the system which are
designed to eliminate unknowns from as many equations as
possible. The important point about these operations is that,
although they change the linear system, they do not change
its solutions.
We begin by subtracting equation 1 from equations 2 and
3 in order to eliminate x\ from the last two equations. These
operations can be conveniently denoted by (2) — (1) and (3) —
(1) respectively. The effect is to produce a new linear system
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