Question

Let A be an N by N matrix with rank J<N. Prove then that there are N - J linearly independent solutions of the system Ax = 0, and the null space of A has dimension N - J.

Answer 1

Every solution is the solutionto Bx=0, where B is row echelon form. For B we have n-j last zero rows, and
these corresponding x_i 's can be choosen linearly independent, and other will
be automatically computed, so there will be n-j lin indep solutions.