Geometric Interpretation of Inconsistency
Learn to identify inconsistent linear systems through their geometric interpretations. This lesson covers how parallelism and non-coincident intersections among lines and planes cause no solutions in two and three dimensions, helping you visualize and understand inconsistency in linear algebra.
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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
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
Planes
The same phenomenon can be extended to planes. The linear system below represents two parallel planes that never intersect, therefore possessing no solution.
Non-coincident intersection
Other than parallelism, inconsistency also arises when the entities defined by the equations in the linear system fail to intersect coincidently.
Lines
In the system below, no pair of lines are parallel, but they fail to intersect at a common point.
We can regenerate the plot above by running the following code. We can also change the coefficients to experiment with the plot.
Planes
Similarly, a three-dimensional linear system fails to possess a solution if the planes represented by the equations of the system fail to intersect at a common point (unique solution) or at a line (infinite solutions). Below is an example of such an inconsistent linear system in which each pair of planes meet but fail to intersect at a common point or line.