Motivated by a connection with block iterative methods for solving linear systems over finite fields, we consider the probability that the Krylov space generated by a fixed linear mapping and a random set of elements in a vector space over a finite field equals the space itself. We obtain an exact formula for this probability and from it we derive good lower bounds that approach 1 exponentially fast as the size of the set increases.
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Brent et al. (2003) studied this question.
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