We study the vectors related tro states on the Cuntz algebra Ο(sub)n and prove hat, for tow states $omega$ and $ ho$ on Ο(sub)n with $omega$│UHF(sub)n = $ ho$│UHF(sub)n, if ($omega$(s$_1$), …, $omega$(s(sub)n)) and ($ ho$(s$_1$),…, $ ho$(s(sub)n)) are unit vectors, then they and linearly dependent. We also study the linear functional on Ο(sub)n associated to a sequence of unit vectors in C(sup)n which is the generalization of the Cuntz state. We show that if the linear functional associated to a sequence of unit vectors with a certain condition is a state, then it is just the Cuntz state.