A note on preserving the spark of a matrix

Abstract

Let $M_{m×n}(F)$ be the vector space of all m×n matrices over a field $F$. In the case where $m ≥ n, char(F) ≠ 2$ and $F$ has at least five elements, we give a complete characterization of linear maps $Φ: M_{m×n}(F) -> M_{m×n}(F)$ such that $spark(Φ(A)) = spark(A)$ for any $A ∈ M_{m×n}(F)$.