Coplanarity

Q:  Given a set of n-dimensional vectors, is there a simple way to check if they are coplanar in ?

A: Eckhard Hennig (aidev@kaninkolo.de) answers: To establish coplanarity in of a set of -dimensional vectors, , you need to check whether the dimension of the null space (or nullity) of the linear mapping defined by has dimension . Here is an example.

Define three linearly independent vectors in .

Generate a set of 10 coplanar vectors in by forming random linear combinations of and . Use SeedRandom to obtain a repeatable sequence of pseudorandom numbers.

The null space of the mapping defined by has dimension 1; hence, the set is coplanar.

In Version 5, you can use MatrixRank instead to calculate the dimension of the subspace spanned by . The vectors are coplanar if the rank of is 2 (or in the general case).

NullSpace may be unreliable for large and ill-conditioned systems. A more reliable, but less efficient, alternative is to determine the number of singular values of the mapping.

Since there are two singular values, the set of vectors spans a two-dimensional space, that is, the vectors are coplanar in .

Now crosscheck for the noncoplanar case: generate 10 vectors in by forming random linear combinations of the three independent basis vectors , , and .

The null space is empty, which implies that , , and span .

Alternatively, the number of singular values equals the dimension of the space.