Question

Let p(x) = a0 + a1x + a2x2 and q(x) = b0 + b1x + b2x2 be vectors in P2 with p, q = a0b0 + a1b1 + a2b2. Determine whether the polynomials form an orthonormal set, and if not, apply the Gram-Schmidt orthonormalization process to form an orthonormal set. (If the set is orthonormal, enter ORTHONORMAL in both answer blanks.)

{−2 + x², −2 + x}

u1=

u2=

Answer 1

Let p(x) = −2 + x² and q(x) = −2 + x. Then < p, q > = (-2)*(-2)+0*1+1*0 = 4 so that the vectors p(x) and q(x) are not orthogonal.

Let w₁= (-2,0,1) and w₂ = (-2,1,0). Also let v₁ = w₁= (-2,0,1) and v₂ = w₂- proj_v1 (w₂) = w₂- [(w₂.v₁)/(v₁.v₁)]v₁ = w₂- [(4+0+0)/(4+0+1)]v₁ = (-2,1,0) –(4/5) (-2,0,1) = (-2/5, 1,-4/5). Then {v₁,v₂₂} is an orthogonal set.

Further, let e₁ = v₁/||v₁|| = (-2,0,1)/√(4+0+1) = (-2/√5,0,1/√5) and e₂ = v₂/||v₂|| =(-2/5, 1,-4/5)/√( 4/25+1+16/25) = (-2/3√5, √5/3, -4/3√5).

Then { e₁, e₂} is an orthonormal set.

Finally, u₁ = -2/√5 +x²/√5 and u₂= -2/3√5+ x√5/3 -4x²/3√5 .