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Let V be an (n + l) -dimensional vector space over the finite field F q with l ≥ n > 0 and W be a fixed l -dimensional subspace of V. We say that an m -dimensional subspace U of V is of type (m , k) if dim (U ∩ W) = k. Denote the set of all subspaces of type (m , k) in V by M (m , k ; n + l , n). The collection of all the subspaces of types (m , 0) in V , where 0 ≤ m ≤ n , is the attenuated space. In this paper, we prove the Hilton–Milner theorem for M (m , 0 ; n + l , n) , where 3 ≤ m ≤ n. [ABSTRACT FROM AUTHOR]