▁is ▁still ▁considered ▁a ▁linear ▁model ▁because ▁it ▁is ▁linear ▁in ▁the ▁ β s . ▁ ▁Matrix / vector ▁form ulation ▁Consider ▁an ▁over det erm ined ▁system ▁ ▁of ▁n ▁linear ▁equations ▁in ▁p ▁unknown ▁coefficients , ▁ β 1 , β 2 , … , β p , ▁with ▁n ▁> ▁p . ▁( Note : ▁for ▁a ▁linear ▁model ▁as ▁above , ▁not ▁all ▁of ▁ ▁contains ▁information ▁on ▁the ▁data ▁points . ▁The ▁first ▁column ▁is ▁populated ▁with ▁ones , ▁, ▁only ▁the ▁other ▁columns ▁contain ▁actual ▁data , ▁so ▁here ▁p ▁= ▁number ▁of ▁re gress ors ▁+ ▁ 1 .) ▁This ▁can ▁be ▁written ▁in ▁matrix ▁form ▁as ▁ ▁where ▁ ▁Such ▁a ▁system ▁usually ▁has ▁no ▁exact ▁solution , ▁so ▁the ▁goal ▁is ▁instead ▁to ▁find ▁the ▁coefficients ▁ ▁which ▁fit ▁the ▁equations ▁" best ", ▁in ▁the ▁sense ▁of ▁solving ▁the ▁quadratic ▁minim ization ▁problem ▁ ▁where ▁the ▁objective ▁function ▁S ▁is ▁given ▁by ▁ ▁A ▁just ification ▁for ▁choosing ▁this ▁criter ion ▁is ▁given ▁in ▁Properties ▁below . ▁This ▁minim ization ▁problem ▁has ▁a ▁unique ▁solution , ▁provided ▁that ▁the ▁p ▁columns ▁of ▁the ▁matrix ▁ ▁are ▁linear ly ▁independent , ▁given ▁by ▁solving ▁the ▁normal ▁equations ▁ ▁The ▁matrix ▁ ▁is ▁known ▁as ▁the ▁moment ▁matrix ▁of ▁re gress and ▁by ▁re gress ors . ▁Finally , ▁ ▁is ▁the ▁coefficient ▁vector ▁of ▁the ▁least - squ ares ▁hyper plane , ▁expressed ▁as ▁▁ ▁Est imation ▁▁ ▁Suppose ▁b ▁is ▁a ▁"