Final answer:
The Gram-Schmidt process is used to create an orthonormal basis from the polynomials 1, x, x². This involves orthogonalizing and normalizing each polynomial in the sequence with respect to the given inner product.
Step-by-step explanation:
The question asks us to apply the Gram-Schmidt process to the powers 1, x, x², ... to find the first three orthogonal polynomials and then normalize them to get an orthonormal basis for the space of polynomials with the inner product defined by (f,g)=∫¹₀ f(x)g(x)xdx. The Gram-Schmidt Process involves taking a set of linearly independent vectors and, through a process of orthogonalization followed by normalization, producing an orthonormal set of vectors.
Starting with the sequence 1, x, x², we first treat these as our starting vectors. To apply Gram-Schmidt, we begin by setting our first vector p1(x) = 1. Then, subtract from x its projection onto p1 to get a vector orthogonal to p1, which we'll call p2. Normalize p1 and p2 to get the first two orthonormal vectors. Continue with x², subtracting projections onto the previous polynomials and normalizing to get the third vector.
Without going through all the calculations, the general formula for the nth step in Gram-Schmidt is:
pn(x) = xn - ∑¹⁴ pi(x) · (xn, pi(x)) / (pi(x), pi(x))
Normalize each pn(x) to get an orthonormal set.