Question

Apply the Gram-Schmidt process to the vectors [1; −2; 0], [1; 0; −1] and [0; 1; 1]  .

Answer 1

Let v₁ = (1,-2,0),v₂ = (1,0,-1) and v₃= (0,1,1).

Also, let u₁ = v₁ = (1,-2,0), u₂ = v₂ -proj_u1(v₂) = v₂ –[(v₂.u₁)/(u₁.u₁)]u₁ = v₂ –[(1+0+0)/(1+4+0)]u₁= (1,0,-1)-(1/5) (1,-2,0) = (4/5,2/5,-1) and

u₂ = v₃ -proj_u1(v₃)- proj_u2(v₃) = v₃ -[(v₃.u₁)/(u₁.u₁)]u₁- [(v₃.u₂)/(u₂.u₂)]u₂ = v₃ -[(0-2+0)/(1+4+0)]u₁ –[(0+2/5-1)/(16/25+4/25+1)]u₂ = (0,1,1)+(2/5)(1,-2,0)+(1/3) (4/5,2/5,-1)= (2/3,1/3,2/3).

Then {u₁,u₂,u₃} = {1,-2,0), (4/5,2/5,-1), (2/3,1/3,2/3)} is an orthogonal set.

Now, let w₁ = u₁/||u₁|| = (1/√5,-2/√5,0), w₂ = u₂/||u₂|| =(4/5,2/5,-1)= (4/3√5, 2/3√5,-√5/3) and w₃ = u₃=(2/3,1/3,2/3).

Then { w₁, w₂, w₃}= {(1/√5,-2/√5,0), (4/3√5, 2/3√5,-√5/3), (2/3,1/3,2/3)} is an orthonormal set.