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Assume that we have a set of vectors with cardinality $n$ in a dimension $d$ vector space. I need to check if there exists any number of $d$ vectors in the set which is linearly dependent. Is there any more efficient way than checking the linear dependence of all possible $d$-tuple vectors?

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  • $\begingroup$ If you're doing calculations explicitly, it is helpful to note that if you write the $d$ vectors as a $d \times d$ matrix, its determinant is zero if and only if the vectors are linearly dependent. See here for more. $\endgroup$ – Bill Wallis May 4 '18 at 12:29

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