SOLUTION: Let U be a subspace of R^4 and let the set S={x1,x2,x3} be an orthogonal basis of U, where x1=[1,0,-1,-1] x2=[2,1,1,1] x3=[-1,3,-1,0] a) find a basis of U^(orthogonal symbol)

Question 375894: Let U be a subspace of R^4 and let the set S={x1,x2,x3} be an orthogonal basis of U, where x1=[1,0,-1,-1] x2=[2,1,1,1] x3=[-1,3,-1,0]
a) find a basis of U^(orthogonal symbol)
b) Given x=[3,1,0,42] find vectors x1 and x2 such that x= x1+x2, where x1 is in U and x2 is in U^(orthogonal symbol)
Answer by Jk22(389) (Show Source): You can put this solution on YOUR website!
a) We take a vector which is orthogonal to : x1=[1,0,-1,-1] x2=[2,1,1,1] x3=[-1,3,-1,0]
let u[a,b,c,d], u.x1=u.x2=u.x3=0

a-c-d=0
2a+b+c+d=0
-a+3b-c=0
-a+3b-c=0
3b-2c-d=0
7b-c+d=0
-a+3b-c=0
3b-2c+d=0
10b-3c=0

3c=10b
d=3b-2*10b/3=-11b/3
a=3b-c=3b-10b/3=-b/3

u=[-1,3,10,-11]

since it's of dimension 1 (x1,x2,x3 are linearly independent), another way would be to calculate the cross product of x1,x2,x3 :
[e1 1 2 -1]
[e2 0 1 3]
[e3 -1 1 -1]
[e4 -1 1 0]
=e1
[0 1 3]
[-1 1 -1]
[-1 1 0]

-e2
[1 2 -1]
[-1 1 -1]
[-1 1 0]

+e3
[1 2 -1]
[0 1 3]
[-1 1 0]

-e4
[1 2 -1]
[0 1 3]
[-1 1 -1]

=e1(-(-1)*(1*0-1*3)-1*(1*(-1-3)))
-e2(-1*(-2+1)-1*(-1-1))
+e3(-3-7)
-e4(-4-(7))

=[1,-3,-10,11]=u1=-u
b) Given x=[3,1,0,42] find vectors x1 and x2 such that x= x1+x2, where x1 is in U and x2 is in U^(orthogonal symbol)
we project x on u1 :
fact=x.u1/|u1|^2=(3-3+42*11)/Sqrt(1+9+100+121)=462/231=2
x2=fact*u1=[1,-3,-10,11]*2=[2,-6,-20,22]
x1=x-x2=[3,1,0,42]-[2,-6,-20,22]=[1,7,20,20]

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