U=span{[1 -2 1 -1], [ 2 1 -1 1]}. show that Y=[1 3 -2 2] is in U, and find all Z such that [Y,Z] is an orthogonal basis of U.
Notice that {[1 -2 1 -1], [ 2 1 -1 1]} is an orthogonal basis for U.Y = [2 1 -1 1] - [1 -2 1 -1]
Writing Y with respect to this basis is (-1,1)
so (-1,1)$\displaystyle \bullet$ Z = 0
so Z can be span{(1,1)}
so Z can be span {(3,-1,0,0)}