Apply the GramSchmidt orthonormalization process to transfor

Apply the Gram-Schmidt orthonormalization process to transform the given basis for R^n into an orthonormal basis. Use the vectors in the order in which they are given. B = {(8, 15), (1, 0)} u_1 = (8/17, 15/17) u_2 =

Solution

Let v₁ = (8,15) and v₂ = (1,0). Also, ley u₁ = v₁ = (8,15) and u₂ = v₂- proj_u1 (v)= v₂-[(v₂.u₁)/(u₁.u₁)]u₁ = v₂-[(8+0)/(64+225)]u₁=(1,0)–(8/289)(8,15)=(225/289, -120/289).Then {u₁,u₂} ={(8,15),(225/289, -120/289)} is an orthogonal basis for R². For determining an orthonormal basis, we need to convert u₁,u₂ into unit vectors. Let e₁ = u₁/||u₁|| = u₁/(8²+15²) = (1/289)u₁ = (1/17(8,15) = (8/17,15/17) and e₂ = u₂/||u₂|| = u₂/[(225/289)²+(-120/289)²]=u₂/(50625/83521+14400/83521)=u₂/65025/83521)=[1/(255/289)]u₂= (289/255)(225/289, -120/289) = ( 225/255, -120/255) = (45/51,-8/17).

Then, {e₁,e₂} = {(8/17,15/17), (45/51,-8/17)} is the required orthonormal basis for R².