Question

a) Verify that B is a basis for IR3 (b) Use the Gram-Schmidt process to produce an orthogonal basis for R (c) Normalize the vectors to produce an orthonormal basis for R3.

Answer 1

2.(a). Let v₁ = (1,-1,0)^T, v₂ = (2,0,1)^T and v₃ = (5,1,0)^T.Let A be the matrix with the given vectors as columns. To determine whether the vectors in B are linearly independent and span R³, we will reduce A toits RREF which is I₃. It implies that the given vectors are linearly independent and span R³ so that B is a basis for R³.

(b). Let u₁=v₁=(1,-1,0)^T, u₂=v₂–proj_u1(v₂)= v₂–[(v₂.u₁)/(u₁.u₁)]u₁ = v₂–[(2+0+0)/(1+1+0)]u₁ =(2,0,1)^T -(1,-1,0)^T = (1,1,1)^T and u₃=v₃–proj_u1(v₃)- proj_u2(v₃)= v₃–[(v₃.u₁)/(u₁.u₁)]u₁-[(v₃.u₂)/(u₂.u₂)]u₂ = v₃–[(5-1+0)/(1+1+0)]u₁ –[ (5+1+0)/(1+1+1)]u₂ =(5,1,0)^T -2(1,-1,0)^T-2(1,1,1)^T =(1,1,-2)^T.Then {u₁,u₂,u₃} is an orthogonal basis for R³.

Now, let e₁ = u₁/||u₁||= (1/?2,- 1/?2,0)^T, e₂ = u₂/||u₂||= (1/?3,1/?3,1/?3)^T and e₃ = u₃/||u₃||= (1/?6,1?6,-2/?6)^T. Then {e₁,e₂,e₃} = {(1/?2,- 1/?2,0)^T, (1/?3,1/?3,1/?3)^T , (1/?6,1?6,-2/?6)^T } is an orthonormal basis for R³.