Geometric Interpretation of Inconsistency

Inconsistent linear system

An inconsistent linear system is a linear system that doesn’t possess any solution. As explained previously, we can identify such a linear system by the presence of a pivot in the last column of the $rref$ of its augmented matrix. In this lesson, we’ll present the geometric interpretations of inconsistent linear systems with examples.

Geometrically, the case of no solution in linear systems arises from two possible sources.

Parallelism

No solution exists when a given set of lines, planes, or spaces are parallel to each other. In the following subsection, we present graphical representations of two-dimensional and three-dimensional linear entities, while leaving higher dimensional linear entities to the reader's imagination.

Lines

The linear system above is inconsistent. In particular, there’s no point $(x,y)$ that satisfies both equations. Geometrically, the equations represent lines in the $xy-$ ...