Use the GramSchmidt Procedure to produce an orthogonal basis

Use the Gram-Schmidt Procedure to produce an orthogonal basis for the subspace spanned by the set. {[-1 2 1 1], [7 -8 -1 -4], [5 4 7 -3]} An orthogonal basis for the subspace spanned by the given set is {}. (Use a comma to separate matrices as needed.)

Solution

Let the given vectors be denoted by v₁,v₂and v₃.Then, as per Gram-Schmidt process, u₁ = v₁ = (-1,2,1,1)^T, u₂ = v₂ –proj_u1(v₂) = v₂ –[(v₂.u₁)/(u₁.u₁)]u₁= v₂ –[(-7-16-1-4)/(1+4+1+1)]u₁ = (7,-8,-1,-4)^T + 4(-1,2,1,1)^T = ( 3,0,3,0)^T, and

u₃= v₃- proj_u1(v₃)- proj_u2(v₃)=v₃-[(v₃.u₁)/(u₁.u₁)]u₁-[(v₃.u₂)/(u₂.u₂)]u₂=v₃-[(-5+8+7-3)/(1+4+1+1)]u₁                                 –[(15+0+21+0)/(9+0+9+0)]u₂= (5,4,7,-3)^T-(-1,2,1,1)^T- 2( 3,0,3,0)^T = (0,2,0,-4)^T. Thus an orthogonal basis for the given subspace is { (-1,2,1,1)^T, ( 3,0,3,0)^T, (0,2,0,-4)^T }