▁gradient ▁vector ▁of ▁the ▁motion ▁of ▁the ▁vertices ▁must ▁have ▁zero ▁inner ▁product ▁with ▁the ▁row ▁of ▁the ▁rig id ity ▁matrix ▁that ▁represents ▁the ▁given ▁edge . ▁Thus , ▁the ▁family ▁of ▁grad ients ▁of ▁( inf init es im ally ) ▁rig id ▁mot ions ▁is ▁given ▁by ▁the ▁null space ▁of ▁the ▁rig id ity ▁matrix . ▁For ▁frameworks ▁that ▁are ▁not ▁in ▁generic ▁position , ▁it ▁is ▁possible ▁that ▁some ▁inf init es im ally ▁rig id ▁mot ions ▁( ve ctors ▁in ▁the ▁null space ▁of ▁the ▁rig id ity ▁matrix ) ▁are ▁not ▁the ▁grad ients ▁of ▁any ▁continuous ▁motion , ▁but ▁this ▁cannot ▁happen ▁for ▁generic ▁frameworks . ▁ ▁A ▁rig id ▁motion ▁of ▁the ▁framework ▁is ▁a ▁motion ▁such ▁that , ▁at ▁each ▁point ▁in ▁time , ▁the ▁framework ▁is ▁con gru ent ▁to ▁its ▁original ▁configuration . ▁R ig id ▁mot ions ▁include ▁transl ations ▁and ▁rot ations ▁of ▁E uclidean ▁space ; ▁the ▁grad ients ▁of ▁rig id ▁mot ions ▁form ▁a ▁linear ▁space ▁having ▁the ▁transl ations ▁and ▁rot ations ▁as ▁bases , ▁of ▁dimension ▁, ▁which ▁must ▁always ▁be ▁a ▁sub space ▁of ▁the ▁null space ▁of ▁the ▁rig id ity ▁matrix . ▁Because ▁the ▁null space ▁always ▁has ▁at ▁least ▁this ▁dimension , ▁the ▁rig id ity ▁mat roid ▁can ▁have ▁rank ▁at ▁most ▁, ▁and ▁when ▁it ▁does ▁have ▁this ▁rank ▁the ▁only ▁mot ions ▁that ▁preserve ▁the ▁lengths ▁of ▁the ▁edges ▁of ▁the ▁framework ▁are ▁the ▁rig id