Let b 1 , … , b n {\displaystyle {}b_{1},\ldots ,b_{n}} be a basis of V {\displaystyle {}V} . Due to the basis exchange theorem, there are n − k {\displaystyle {}n-k} vectors from the basis b {\displaystyle {}{\mathfrak {b}}} which together with the given vectors u 1 , … , u k {\displaystyle {}u_{1},\ldots ,u_{k}} form a basis of V {\displaystyle {}V} .
