A set of vectors is k-independent if all its subsets with no more than k elements are linearly independent. We obtain a result concerning the maximal possible cardinality Indq(n, k) of a k-independent set of vectors in the n-dimensional vector space F qnover the finite field F qof order q. Namely, we give a necessary and sufficient condition for Indq(n, k) = n + 1. We conclude with some pertinent remarks re applications of our results to codes, graphs and hypercubes.