Find a basis for the orthogonal complement of the subspace g

Find a basis for the orthogonal complement of the subspace generated by {[2, -1, 0, -3]}.

Solution

Let the given subspace be denoted by W. Also, let X = (x , y, z, w)^T be an arbitrary element of W, the orthogonal complement of W. Then (x , y, z, w)^T. (2,-1,0,-3)^T = 0 or, 2x –y -3w = 0 or, x = ½(y+3w). Let y = 2r, z = s and w = 6t. Then x = r+3t and X = (r+3t, 2r,s,6t)^T = r(1,2,0,0)^T + s(0,0,1,0)^T +t(3,0,0,6)^T. Thus, a basis for W is { (1,2,0,0)^T, (0,0,1,0)^T, (3,0,0,6)^T}