Subsection NSM Null Space of a Matrix

The set of solutions to a homogeneous system (which by Theorem HSC is never empty) is of enough interest to warrant its own name. However, we define it as a property of the coefficient matrix, not as a property of some system of equations.

Definition NSM Null Space of a Matrix

The null space of a matrix $A$, denoted $\nsp{A}$, is the set of all the vectors that are solutions to the homogeneous system $\homosystem{A}$.

In the Archetypes (Archetypes) each example that is a system of equations also has a corresponding homogeneous system of equations listed, and several sample solutions are given. These solutions will be elements of the null space of the coefficient matrix. We will look at one example.

Example NSEAI Null space elements of Archetype I

Here are two (prototypical) examples of the computation of the null space of a matrix.

Example CNS1 Computing a null space, no. 1
Example CNS2 Computing a null space, no. 2
Sage NS Null Space
Sage SH Sage Help
(in context)