Question

xn and B is n x m with AB = Im and n linearly independent and b) the Suppose that A is m Show that a) the columns of B are т. linearly dependent rows of A are

. Suppose that A is m × n and B is n × m with AB = Im and n 6= m. Show that a) the columns of B are linearly independent and b) the rows of A are linearly dependent.

Answer 1

Рузи Бихт AB= Im R(AB)= Rank of (AB) = m R(A)= Rank of A Rank (AB) = min (RIA), RIB)) R(A) = Rank of B ma min & R(A), R(R)} me R(A) < miglm, n} = R(A)= m since nam m = R(B) = mixfm, n} = R(B) = m since nom R(A) =m column nulity of A = nom 3 colums of A are dependent. Row nudity of A=m.m=0 = Rows of A are dependent. and RB=m Column Nellity of B = m-m=0 7 calums are Independent Row Nallity of B = nom - Rows of A are dependent. of B