Elementary Operations on a System of Linear Equations
There are three basic kinds of operations that can be used to solve any
system of linear equations. These operations are simple transformations
of the original system of equations that preserve the
information content about the solution.
The elementary operations are:
Clearly, these elementary transformations preserve the information content
of the original system of equations for the simple reason of being
reversible. In other words, after applying a sequence
of these operations to the original system we'll end up with another
system that may look very different from the initial one, but by
applying the reversed sequence in reverse we can go back
to the begining. Notice that we undo the first kind
of elementary transformation (switching two rows) by doing it again.
That is, the reverse of switching is itself!
Let us call k the number that multiplies the equations in the definition
of the elementary operations of the second and third kind above. We can now see
that the operations that undo them are also of the
second and third kind but with -k instead of k.
OK. But what's so special about these operations?
Well, they are enough to solve ALL possible linear systems of equations.
We'll see that by applying sequences of these transformations we can
always find the solution (if there is a solution) to the system or find
out that there is no solution. A systematic procedure to accomplish
this is Gaussian Elimination.