rank-nullity theorem. In problem 3 write to be the matrix consisting of these columns. You want to show that has a solution for any . Define by , this is a linear transformation. We know that (the dimension of ). The rank of is the dimension of the image of i.e. . Thus, is onto (which is what we want to show) if and only if since the dimension of is . It remains that all we need to show is that . Remember is the dimension of the nullspace of i.e. . Thus, you need to show that the set of solution to has one element in its basis. Now procede by Gaussian-Jordan elimination.