<prompt> Acha mathbfv iwe vekta kama hiyokushoto mathbfv anzapmatrix 3 -1 mwishopmmatrix kulia 8.Tafuta thamani ndogo kabisa ya mathbfv.  <response>: Kwa vekta zote mathbfa na mathbfb, na Kutokuwa na Usawa kwa Pembetatu,mathbfa mathbfb le mathbfa mathbfb.Hasa,kushoto mathbfv anzapmatrix 3 -1 mwishopmmatrix kulia le mathbfv kushoto anzapmatrix 3 -1 mwishopmatrix kulia.Kwa hivyo,mathbfv ge kushoto mathbfv anzapmatrix 3 -1 mwishopmmatrix kulia - kushoto anzapmatrix 3 -1 mwishopmmatrix kulia 8 - sqrt10.Usawa hutokea tunapochukuamathbfv frac8 - sqrt10sqrt10 anzapmatrix 3 -1 endpmatrix frac8sqrt 10 anzapmatrix 3 -1 mwishopmatrix - anzapmatrix 3 -1 mwishopmatrix,kwa hivyo thamani ndogo kabisa ya mathbf v ni 8 - sqrt10. 