If NAT 0 with A elementof Rm times n then give brief justif

If N(A^T) = {0}, with A elementof R^m times n, then give brief justification for your True/False answers to the following statements (answer False if a statement is not guaranteed to be always true!): [] the system Ax = b will have a unique solution. [] The rank of A equals n. [] The rank of A equals m. [] N(A) = {0} [] n greaterthanorequalto m

Solution

If N(A^T) = {0} and A is m x n matrix then we can say that A^T is n x m matrix.

So by rank-nullity theorem,

rank(A^T) + nullity(A^T) = no. of columns of A^T

=> rank(A^T) + 0 = m

=> rank(A^T) = m

We know that rank(A) = rank(A^T)

Therefore rank(A) = m

b) False (Prove above)

c) True (Prove above)

d) False

rank(A) + nullity(A) = no. of columns of A

nullity(A) = n - m

nullity(A) = 0 when n=m and we are not sure about that.

e) True

Because rank(A) min(m,n)

since rank(A) = m which is minimum between the m and n.

Therefore n m

a) False

rank(A) n.

so there are 2 possibilities,

1) m=n then rank(A) = m = n

then we have unique solution.

2) m<n then rank(A) = m < n

then we have infinitely many solution.

So we are not sure.