 Kernel jump in a parameter-dependent linear map

For a matrix whose entries depend on a parameter, a nonzero maximal minor certifies a fixed matrix rank away from its zeros. At a zero, other minors must still be checked: one vanishing minor does not itself imply a rank drop. Once the matrix rank is determined, the rank-nullity theorem gives the dimension of the kernel of a linear map. Exceptional parameters can then be solved separately to exhibit the additional kernel of a linear map directions.