Applications of Linear Systems
Many mathematical and scientific problems can be formulated in terms of
systems of linear equations. Sometimes two or three equations suffice,
but in many cases several dozen or even hundreds or thousands are needed.
Systems with two or three unknowns can be worked out by hand and can easily
be interpreted geometrically (as intersections of lines or planes in space).
Larger systems arise in many mathematical, scientific and engineering problems,
e.g., determining the flow of water through a
network of pipes given certain pressure and flow measurements, construction
of curves that pass through specified points (the interpolation problem)
and determining the temperature at all points of a body at equilibrium
given measured temperatures along its boundary. A very crude form of the
last problem easily leads to systems with hundreds of unknowns. The theory
for dealing with these large systems has long been known: it is the process
of Gaussian elimination, which works by eliminating one variable at a time
to reduce a system of n equations in n
unknowns to
a system of n-1 equations in n-1 unknowns, and so on, until
one arrives at a system of one equation in one unknown, which anyone can
solve. The remaining variables are then found by "back-substitution". To
apply the theory in a practical way requires an electronic assistant, e.g.
MATLAB, to do the vast amount of drudge work.