Thread: orthogonal basis

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.

Originally Posted by akhayoon

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.

Y = [2 1 -1 1] - [1 -2 1 -1]

well, thanks but...what about the rest?

Y = [2 1 -1 1] - [1 -2 1 -1]

Notice that {[1 -2 1 -1], [ 2 1 -1 1]} is an orthogonal basis for U.
Writing Y with respect to this basis is (-1,1)
so (-1,1) Z = 0
so Z can be span{(1,1)}
so Z can be span {(3,-1,0,0)}

so you're proving that Z is orthogonal using the dot product?

I think you need to revise the definition of orthogonal.