= 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.
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