Suppose A is a 7 times 7 matrix with rankA 5 What is the di

Suppose A is a 7 times 7 matrix with rank(A) = 5. What is the dimension of ker(A)? Clearly explain your answer.

Solution

Since the kernel of A and the null space of A have the same dimension, we can use the rank-nullity theorem to deduce that ker(A).

i.e. rank(A)+nullity(A)=n

[i.e. dim(range of A)+dim(kernel of A)=dim(domain of A)]

Therefore, dim(kernel of A)=7-5=2.

Hence, dim(ker(A))=2.