SYSTEMS OF LINEAR EQUATIONS
1. Systems of Linear Equations
Linear equations are by themselves not particularly interesting. More often
than
not, one encounters collections of linear equations, involving the same
variables,
which are to be considered simultaneously.
Definition. An m*n system of linear equations is a system of the form
In particular, there are m equations in the n variables x_{1}, x_{2},
… , x_{n}.
The numbers
a_{ij} are called the coefficients of the system (1) and the b_{i} are called the constant
terms.
Definition. A system of equations that has no solutions is called
inconsistent. If
there is at least one solution of the given system, then it is called
consistent.
In particular, note from the definition that a given linear system of
equations
must be either consistent or inconsistent  there are no other possibilities.
Example 1. The 2*3 system
is consistent. This requires a brief geometric explanation (or a tedious
calculation).
It turns out that the solution set of (2) corresponds to a line in R^{3}. In
particular,
we are asserting that there are infinitely many solutions to the system (2).
Indeed,
the two equations determine two distinct planes in R^{3 }which are not parallel
(their
normal vectors (24, 47, 31) and (67, 55,79) are not parallel since they are
not
scalar multiples of each other). Geometrically, we know that two such planes in
R^{3}
must intersect each other and that this intersection must be a line.
See Figure 3.5.2 (p.157) of Anton which depicts the
numerous ways in which three
planes in R^{3} might intersect. Have a close look at this drawing,
since it illustrates
the geometry of all possible solution sets for 3*3 systems of linear equations.
Example 2. Consider the system
where a, b, c, d, u, v are constants and x, y are the
variables. In terms of the familiar
slopeintercept formula for lines in the plane R^{2}, we can rewrite the
system as
and regard these two equations as defining lines l_{1}
and l_{2}, respectively. It follows
that l_{1} and l_{2} are parallel (or possibly equal) when they
have the same slope
(i.e., a/b = c/d). This is geometric condition is equivalent to the purely
algebraic
condition
We can summarize this as follows:
(i) If ad  bc = 0, then there are two possibilities:
(a) l_{1} and l_{2} are parallel but not
equal (i.e., they are distinct lines having
the same slope). In this case l_{1} and l_{2} do not intersect
and hence the
system (3) has no solutions (the system is therefore inconsistent).
(b) l_{1} and l_{2} are in fact the same
line and there are infinitely many
solutions to the system (3) exist (the system is therefore consistent).
(ii) If adbc ≠ 0, then l_{1} and l_{2}
have different slopes and hence must intersect
and exactly one point, say (x_{0}, y_{0}). Thus the system (3)
has exactly one
solution (the system is therefore consistent).
Of course, we need no geometry whatsoever to consider the
system (3). Indeed,
you have all solved systems consisting of two equations in two unknowns before.
Nevertheless, thinking about things geometrically often helps our intuition and
helps us "picture things." For instance, now it is geometrically clear why the
mysterious quantity ad  bc arises in the consideration of 2*2 linear systems.
For 2*2 systems of the form (3), the quantity ad  bc is
so important that it
has a special name:
Definition. The determinant of a 2*2 system of the
form (3) is defined to be the
real number ad  bc.
We will discuss the determinants of n*n systems in the
near future. However,
for the moment we would like to concentrate on some qualitative aspects of
linear
systems. In particular, we remark that all of our examples have illustrated the
following (see Theorem 1.6.1 of Anton):
Theorem 1. A system of linear equations either has
no solutions, exactly one
solution, or infinitely many solutions.
It is important to note that the preceding theorem only
applies to linear systems
of equations. Indeed, nonlinear systems can have any number of solutions. For
instance, the nonlinear equation x^{2} = 1 (i.e., a system consisting of
one nonlinear
equation in one variable) has two solutions. Systems of linear equations are
quite
special { be careful never to assume that something that works for linear
equations
will work for nonlinear equations.
Example 3. If a linear system has two distinct
solutions, then it must have
infinitely many solutions. For instance, suppose that we have a system of 2345
linear
equations in 874 unknowns. If we can find just two distinct solutions to this
system,
then we can (via the theorem) conclude that the system actually has infinitely
many solutions.
The book has numerous examples (see Section 1.2 of the
text) showing how to
find the solution sets for various systems of linear equations. For this class
you will
rarely be required to solve systems of equations larger than 3*3. On the other
hand, it is important to see and do enough examples to gain a level of
familiarity
with linear systems.
2. Homogeneous Linear Systems
Definition. A system of linear equations is said to
be homogeneous if the constant
terms are all zero. In other words, an m*n (i.e., m equations in n unknowns)
homogeneous system is one of the form
Observe that every homogeneous system is consistent since
the trivial solution
is obviously a solution to (5). Other solutions to the
system (5), if they exist at all,
are referred to as nontrivial solutions.
Since a homogeneous linear system always has at least the
trivial solution, it
follows (from Theorem 1) that exactly one of the following is true for a system
of
the form (5):
(i) The system (5) has only the trivial solution
(ii) The system (5) has infinitely many solutions in addition to the trivial
solution. In other words, the system has
infinitely many nontrivial solutions.
Example 4. Consider the general 2*2 homogeneous
linear system below
The two equations in (6) represent lines l_{1} and
l_{2} in R^{2} which pass through the origin
(0, 0). This corresponds to the fact that a homogeneous system of equations
always
has at least the trivial solution. In fact, the only way that nontrivial
solutions to
(6) can exist is if l_{1} = l_{2}. This is because l_{1}
and l_{2} are guaranteed to meet at the
origin  if they intersect elsewhere, then they must actually be the same line.
Recall that if ad  bc = 0, then l_{1} and l_{2}
have the same slope. Since l_{1} and l_{2}
have the same yintercept (namely y = 0) it follows that ad  bc = 0 means that
l_{1} = l_{2}. In other words:
(i) If adbc = 0, then (6) has infinitely many nontrivial
solutions (in addition
to the trivial solution x = y = 0).
(ii) If adbc ≠ 0, then (6) has exactly one solution,
namely the trivial solution.
The fact that (6) is homogeneous is crucial here. If the
constant terms were
not both zero, then l_{1} and l_{2} could have the same slope
(i.e., ad  bc = 0) yet not
intersect at all (they could be parallel to each other).
Another important fact about homogeneous linear systems is
the following:
Theorem 2. A homogeneous system of linear equations
with more unknowns than
equations must have infinitely many solutions.
It is important to note that the preceding theorem
(Theorem 1.2.1 of the text)
applies only to homogeneous systems (see problem 1.2.28 of Anton).
Example 5. A system of the form
(where a, b, c, d, e, f are constants and x, y, z are the
variables) always has infinitely
many solutions. Indeed, geometrically, the preceding equations represent two
planes
P_{1} and P_{2} in R^{3}. Since the system is
homogeneous, both P_{1} and P_{2} pass through
the origin (0, 0, 0)  in other words P_{1} and P_{2} are
guaranteed to intersect (contrast
this with the inhomogeneous case). This is because the trivial solution
x = y = z = 0
is automatically a solution to both equations in the
system (7). Geometrically, we
can see that either P_{1} = P_{2} or P_{1} and P_{2}
intersect in a straight line. In either case,
there are infinitely many solutions to the system (7).
