Unfortunately till now it was not even known if LTCs need to be non-trivially redundant, i.e., need to have {\em one} linear dependency among its low-weight codewords. In this paper we give the first lower bound of this form, by showing that every positive rate constant query LTC must have linearly many redundant low-weight codewords in its dual. We actually prove the stronger claim that the {\em actual test itself} must use $\Omega(n)$ redundant dual codewords (beyond the minimum number of basis elements required to characterize the code); in other words, non-redundant (in fact, low redundancy) local testing is impossible.