Use the inner product u, v = 2u1v1 + u2v2 in R2 and the Gram-Schmidt orthonormalization process to transform {(2, 1), (−2, −9)} into an orthonormal basis. (Use the vectors in the order in which they are given.)

The Gram-Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space R^n. The process takes a finite, linearly independent set S = {v1, ..., vk} for k ≤ n, and generates an orthogonal set S' = {u1, ..., uk} that spans the same k-dimensional subspace of R^n as S.

Given the set S = {(2, 1), (-2, -9)} and the inner product u, v = 2u1v1 + u2v2, we can apply the Gram-Schmidt process as follows:

1. Start with the first vector in S, v1 = (2, 1). Normalize it to get the first vector in S', u1 = v1 / ||v1||. Here, ||v1|| is the norm of v1, which is sqrt[(22 + 11)] = sqrt[5]. So, u1 = (2/sqrt[5], 1/sqrt[5]).

2. Next, subtract the projection of the second vector v2 = (-2, -9) onto u1 from v2 to get a vector orthogonal to u1. The projection of v2 onto u1 is ((u1, v2) / ||u1||^2) * u1 = ((2*-2 + 1*-9) / 5) * u1 = (-4/5, -9/5). Subtracting this from v2 gives v2' = v2 - ((u1, v2) / ||u1||^2) * u1 = (-2 - (-4/5), -9 - (-9/5)) = (-6/5, -36/5).

3. Normalize v2' to get the second vector in S', u2 = v2' / ||v2'||. Here, ||v2'|| is the norm of v2', which is sqrt[(-6/5)^2 + (-36/5)^2] = sqrt[72/5]. So, u2 = (-6/5sqrt[72/5], -36/5sqrt[72/5]).

So, the orthonormal basis for S under the given inner product is {(2/sqrt[5], 1/sqrt[5]), (-6/5sqrt[72/5], -36/5sqrt[72/5])}.