Use the GramSchmidt orthonormalization process to transform

Use the Gram-Schmidt orthonormalization process to transform the basis B = {(0, 6, 12), (12, 0, 0), (1, 1, 1)} for R^3 into an orthonormal basis. Use the Euclidean inner product for R^3 and use the vectors in the order in which they are shown.

Solution

11. Let v₁= (0,6,12),v₂ = (12,0,0)and v₃ = (1,1,1). Also, let u₁= v₁= (0,6,12), u₂ = v₂-proj_u1 (v₂) =                          v₂- [(v₂.u₁)/(u₁.u₁)]u₁= v₂-[(0+0+0)/((0+36+144)]u₁ = v₂ = (12,0,0), and u₃= v₃-proj_u1 (v₃)- proj_u2 (v₃)=                v₃-[(v₃.u₁)/(u₁.u₁)]u_1-[(v₃.u₂)/(u₂.u₂)]u₂= v₃-[((0+6+12)/(36+144)]u₁-[((12+0+0)/(144+0+0)]u₂= (1,1,1)-(18/180)(0,6,12)-(12/144)(12,0,0)= (1,1,1)- (1/10)(0,6,12)-(1/12)(12,0,0) = (1,1,1)-(0,3/5,6/5)-(1,0,0)=               (0,2/5,-1/5). Then {u₁,u₂,u₃} is an orthogonal basis for R³.

Further, let e₁ = u₁ /||u₁||= [1/(0+36+144)] (0,6,12)= (1/65) (0,6,12)= (0, 1/5,2/5).

e₂ = u₂ /||u₂||= (1/144)(12,0,0)= (1/12)(12,0,0)= (1,0,0) and

e₃ = u₃=(0,2/5,-1/5) is already a unit vector.

Then {e₁,e₂,e₃} is an orthonormal basis for R³.