# 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)

Algebra ->  College  -> Linear Algebra -> 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)       Log On

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 Click here to see ALL problems on Linear Algebra 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]