The null space, or kernel, of a m x n matrix A is the set (vector space) of all solutions x to the equation Ax = 0 (including x = 0). The dimension of the null space is r - n, where r is the rank of A
If the null space of A is more than the set {0}, then the columns of A are dependent and A is not invertible. If the null space is {0} (and A is square), the columns are independent, and the matrix is surely invertible.