Let A be an m times n matrix with rank A n Let v w elemento

Let A be an m times n matrix with rank (A) = n, Let v, w elementof R^m with Av = Aw. Does this imply v = w? What happens if rank (A)

Solution

By rank nullity theorem

rank(A)+nullity(A)=n

SO if rank(A)=n

THen, nullity(A)=0

So ker(A)={0}

Hence, if Av=Aw

A(v-w)=0

Then, v-w=0 or v=w

But if rank(A)<n

Then, nullity(A)>0

ANd hence ker(A) is non zero and Av=Aw does not imply v=w