Let A be some m*n matrix. Consider the set S = {z : Az = 0}.First show that this is a vector space. Now show that n = p+q wherep = rank(A) and q = dim(S). Here is how to do it. Let the vectorsx1, . . . , xp be such that Ax1, .. . ,Axp form a basis of the column space of A (thuseach x can be chosen to be some unit vector with a 1 correspondingto the position of a column vector that is part of a (maximally)linearly independent set) and let the vectors z1, . . . , zq be abasis for S. Then show that the two sets together i.e. the set {x1,. . . , xp, z1, . . . , zq} form a basis of the n-dimensionalEuclidean space.
Using the result above, offer a direct proof of the resultr(X?X) = r(X) without appealing to the product rank theorem
