Help with this question?

Let #P_3# be the inner product space of polynomials of degree at most 3 over #R# with respect to the inner product #< f,g> =∫_0^1f(x)g(x)dx#. Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of #P_3# generated by the vectors #{1-2x,2x+6x²,-3x²+4x³}#.?

1 Answer
Cesareo R.
Jan 27, 2018

Given

#u_1 = 1-2x#
#u_2 = 2x+6x^2#
#u_3 = -3x^2+4x^3#

The so called Gram-Schmidt orthogonalization process

#v_1 = u_1#

#v_2 = u_2 - < u_2, v_1 > v_1/norm(v_1)^2#
#v_3 = u_3 - < u_3, v_1 > v_1/norm(v_1)^2- < u_3, v_2 > v_2/norm(v_2)^2#

After that, the normalization

#hat v_1 = v_1/norm(v_1) = sqrt3(1-2x)#
#hat v_2 = v_2/norm(v_2) = sqrt(5/46)(4 - 6 x + 6 x^2)#
#hat v_3 = v_3/norm(v_3)=20 sqrt[161/373] (47/460 - (87 x)/230 - (12 x^2)/23 + x^3)#

Related questions
Impact of this question
334 views around the world
You can reuse this answer
Creative Commons License