292 Larger Linear Systems
- Three different planes, such that each pair of planes intersects in a different line, and
all three of those lines are mutually parallel. - Three different planes that are mutually parallel.
- Two planes that precisely coincide, and a third plane parallel to them both.
- Two planes that precisely coincide, and a third plane that intersects them both in a
single line. - Three different planes that all intersect in a single line.
- Three planes that all precisely coincide.
In the first case, the system has a unique solution, corresponding to the point where all
three planes intersect. In each of the second through fifth cases, there is no solution. In each
of the last three cases, there are infinitely many solutions. Try to envision all these situations.
Note that two planes are parallel in space if and only if they do not intersect.Are you confused?
Most people have trouble envisioning the graphs of three-by-three linear systems. Computer programs
have been developed to portray systems like these, and such programs can be a big help. But they, too,
give only a limited perspective. A true view would require a three-dimensional hologram that we could
walk around in! When it comes to n-by-n linear systems where n is a natural number larger than 3, even a
walk-through hologram can’t give us a complete picture.
It’s not easy to verbally describe what happens in n-by-n linear systems when n is large. Often, a unique
solution exists in this type of system, but not always. A unique solution always comes down to a single
point in Cartesian n-space. When n is large, there are many ways that an n-by-n linear system can fall short
of a unique solution. Even so, if we write up a system of n linear equations in n variables “at random,” the
chance is good that it will have a single solution.
As you can imagine, the process of solving an n-by-n system of linear equations where n > 3 is bound
to be time-consuming and tedious. See how much longer it took us to solve the three-by-three system in
this chapter than it took us to solve the two-by-two systems in Chap. 16! As n increases, so does the time
it will take us to solve the system, unless we have access to a computer. The process of solving an n-by-n
linear system is ideally suited to computer applications, which grind out solutions by brute force.Fewer equations than variables
All of the linear systems we’ve examined so far have the same number of equations as variables.
Occasionally, a linear system has fewer equations than variables. Whenever that happens,
there is no single solution to the whole system.
Think of a linear system with two variables but only one equation. If we consider this as a
one-by-two linear system (one equation, two variables), then it has infinitely many solutions.
It is the same thing as a redundant two-by-two system. In the Cartesian plane, it shows up as
a single straight line.
Now imagine a one-by-three linear system (a single equation in x,y, and z). Its graph in
Cartesian three-space is a single flat plane. The solutions are all the ordered triples (x,y,z) that
represent points on the plane. There are infinitely many such points, so this type of system has
infinitely many solutions. This type of system can never have a unique solution.