Null–space construction using cofactors from a screw–algebra context

Abstract

This paper develops an approach for constructing the null space N ( A ) of a linear system of homogeneous equations using the cofactors of an augmented coefficient matrix A . The relationship between the row space R ( A T ) and null space is exploited by introducing an augmenting vector which is linearly independent of the row space and dependent on the null space. The resultant null space is shown to be a vector of cofactors of the augmenting row of the coefficient matrix and is invariant. This provides a straightforward solution to a linear system of homogeneous equations without going through Gauss-Seidel elimination. The approach is derived from a onedimensional null space and is extended to a multidimensional one by partitioning the coefficient matrix and consequently constructing a set of ( n − m ) null–space vectors based on cofactors. Examples are given and accuracy is compared with Gauss–Seidel elimination. The approach is further used in a screw–algebra context with a simple procedure to obtain a system of reciprocal screws representing a set of constraint wrenches from a set of twists of freedom, in the form of a linear system of homogeneous equations in R 6 . The paper provides rigorous proofs and applications in both linear algebra and advanced kinematics.