Let A be a m × n matrix with entries in R. Recall that the rowrank of A means the dimension of the subspace in RNspanned by the rows of A (viewed as vectors in Rn), andthe column rank means that of the subspace in Rm spannedby the columns of A (viewed as vectors in Rm).
(a) Prove that
n = (column rank of A) + dim S,
where the set S is the solution space of the homogeneousequation AX = 0, that is, S = {column vectors X : AX = 0} .
(b) Show that
row rank of A = column rank of A.
