My overall goal is to find a linear transformation which takes R^3 -> R^3, taking the inner product space with squared norm x^2 + 2xy + 4y^2 + 8z^2 into the std. inner product.
Now, I know how to use the gram-schmidt method to orthonormalize a basis, but I am just completely blanking on how to find a basis to orthonormalize! Any ideas would be great. I don't need the problem solved, just the right nudge :)