Define a vector-valued function $\eta: R^2 \to R^3$ by $\eta_1(x,y)=x^2$, $\eta _2(x,y)=y^2$ and $\eta_3(x,y)=(x-y)^2$, and let $\Omega_0=\eta((0,\infty)^2)$. Let V be the tangent space for $\Omega_0$ at (4,1,1). Find orthonormal vectors that span V.
This is a question from large sample theory that requires a lot of algebra. The book does not explain how to find orthonormal vectors for tangent spaces. Any help is greatly appreciated :)
Find $(x,y)$ such that $\eta((x,y)) = (4,4,1)$ (there are two such points).
Compute ${\partial \eta ((x,y)) \over \partial z}$ and find an orthonormal basis for the range.
This can be done using the Gram Schmidt process.