Linear Systems

• Level One
  • Level Two

***********************

A system is a group of interacting, interrelated, or interdependent elements forming or regarded as forming a collective entity. A set of linear equations composing a system describes and defines the various dimensions modeled by each individual equation of the set; and the solution set of this whole represents the common values satisfying each seperate equation of the collective where they intersect. "Intersection" is a geometric idea carried over into algebra by set theory, the algebra of George Boole. He wrote that the essential characteristic of mathematics is not so much its content as its form; and that mathematics itself is a method resting upon the employment of symbols and precise rules of operation upon these symbols, subject only to the requirement of inner consistency.

Level One

Given a system of linear equations: if they do not intersect at any point, then there is no simultaneous or mutual solution and they are considered to be inconsistent; if there is a single member in the solution set, the system is said to have a unique solution; and, if the members of the set of equations are in actuality one and the same, they are said to have an infinite number of solutions, the set is infinite.

Two systems of equations (which may contain only one equation each) are said to be equivalent if their solution sets are the same. This statement sounds very simple but, in point of fact, it has profound implications. If we have two very dissimilar contextual patterns of relationship having nothing whatever in common apparent on the surface, but which under mathematical analysis prove to have congruent or equivalent underlying structural arrangements, their inner workings, then these two systems or patterns, forms, are expressions of one and the same fundamental rules of relationship. They are said to be isomorphic. We use this deceptively simple idea to apply method to the solution of systems:

Any operation we can perform on the equations of a system that doesn't change its solution set {leaves the solution set 'invariant'} is known as an elementary operation.

There are three such operations commonly used

• Eliminate any equation that has no nonzero coefficient{s};
• Multiply both members of an equation by the same nonzero number;
• Add any real multiple of the members of an equation to the corresponding members of another equation of the system.

For those familiar with the algebra of "Matrices," you might see a similarity, isomorphism, between the above possible operations and those allowed on matrices when reducing to what's called "Row Escelon Form," the basis of the array.

From a geometric point of view, application of these transformation operations leaves the defining underlying features invariant.

Examples