Abstract
A characterization is given of all nonsingular linear operators, on the set of m× n matrices over any field with at least four elements, which map the set of rank- k matrices into itself. It is also shown that if L is any subspace of m× n matrices over any field with at least k+1 elements whose nonzero elements all have rank k, then the dimension of L is at most max( m,n). This fact is used to characterize all linear operators on the set of m× n matrices over certain fields which map the set of rank- k matrices into itself.
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