Let A be an m x n matrix In each of the cases n m m n stat

Let A be an m x n matrix. In each of the cases n > m, m > n, state whether the following are True or False. Give reasons for your answers. There always exists an x notequalto such that Ax = 0. It is possible for the columns of A to span R^m.

Solution

Case(1):- n > m

If A is an m by n matrix then Rank(A) = min{m,n}=m

Rank(A) =m and the number of unknowns are n , n> m hence AX=0 has a non trivial solution

dim(R^m)=m= Rank(A) hence the column space of A spans R^m

Case(2):- m > n

If A is an m by n matrix then Rank(A) = min{m,n}=n

Rank(A) =n and the number of unknowns are n hence AX=0 has trivial solution(X=0 is the solution)

dim(R^m)=m, Rank(A)=n and m > n hence the column space of A not spans R^m

^{Eg: Let {(1,0)} be the columan space of A hence Rank(A) =1. Here the column space of A not spance} R²