  One of the motivations to state HRT conjecture on the linear independence of
finite Gabor systems was the fact that there are linearly dependent Finite
Wavelet Systems (FWS). Meanwhile, there are also many examples of linearly
independent FWS, some of which are presented in this paper. We prove the linear
independence of every three point FWS generated by a nonzero Schwartz function
and with any number of points if the FWS is generated by a nonzero Schwartz
function, for which the absolute value of the Fourier transform is decreasing
at infinity. We also prove the linear independence of any FWS generated by a
nonzero square integrable function, for which the Fourier transform has certain
behavior at infinity. Such a function can be any square integrable function
that is a linear complex combination of real valued rational and exponential
functions.
