Homework SourseFind the orthogonal complement of W Span 0 0 1 1 0 1 1 1Sol
Find the orthogonal complement of W = Span {[0 0 -1 1], [0 1 1 1]}.![Find the orthogonal complement of W = Span {[0 0 -1 1], [0 1 1 1]}.SolutionLet u = (x,y,z,w)T be an arbitrary element of W , the orthogonal complement of W. Th](/WebImages/34/find-the-orthogonal-complement-of-w-span-0-0-1-1-0-1-1-1sol-1101039-1761581747-0.webp "Find the orthogonal complement of W = Span {[0 0 -1 1], [0 1 1 1]}.SolutionLet u = (x,y,z,w)T be an arbitrary element of W , the orthogonal complement of W. Th")
Solution
Let u = (x,y,z,w)T be an arbitrary element of W , the orthogonal complement of W. Then, by the definition of orthogonal complement, we have (x,y,z,w). (0,0,-1,1) = 0 or, -z+w = 0 so that z = w, and (x,y,z,w).(0,1,1,1) = 0 or, y+z+w = 0 or, y +2w = 0(as z=w) or, y = -2w. Then u = (x, -2w,w,w)T = x(1,0,0,0)T + w( 0,-2,1,1)T. Hence, W =span{(1,0,0,0)T, ( 0,-2,1,1)T }.
