11.6 NULLSPACE OF DEPENDENCE MATRIX: THE BROADCAST SUBDOMAIN B
We will see in this section that the nullvector of the dependence matrix A of some variable v describes a subdomain B ⊂ . We will prove that all points in B contain the same instance of v.
11.6.1 The Nullspace of A
If the dependence matrix A is rank deficient, then the number of independent nullvectors associated with A is given by
where n is the number of indices of the algorithm. These nullvectors define the nullspace of matrix A.
Now assume a specific instance for the variable v(c). The following theorem identifies the points in that use that variable.
Consider a particular nullvector e associated with a variable v. If two distinct points p1 and p2 use the same instance v(c), then the vector connecting the two points is a nullvector of A.
Assume the two points use the same instance v(c). Substitute the two points into Eq. 11.21 to get
Subtracting the two equations, ...