linearset

A linearset is a mathematical object that represents a collection of points satisfying a linear equation or a system of linear equations. In simpler terms, it is the set of all possible solutions to one or more linear equations.

The simplest linearset is defined by a single linear equation in one variable, such as ax = b, where 'a' and 'b' are constants and 'x' is the variable. The linearset in this case would be the set containing the single value x = b/a, provided 'a' is not zero. If 'a' is zero and 'b' is also zero, the linearset is all real numbers. If 'a' is zero and 'b' is non-zero, there are no solutions, and the linearset is empty.

In two variables, a linear equation like ax + by = c represents a straight line in a two-dimensional plane. The linearset is therefore the infinite set of all points (x, y) that lie on this line. Similarly, in three variables, a linear equation represents a plane, and the linearset is the set of all points (x, y, z) on that plane.

When dealing with a system of multiple linear equations, the linearset is the intersection of the solution sets of each individual equation. This intersection can be a single point, a line, a plane, or a higher-dimensional analogue, or it can be empty if the equations are inconsistent. Analyzing linearsets is fundamental to understanding linear algebra and solving systems of equations.