Coordinates of a vector under a basis in a Hilbert space?

Question

Given an arbitrary basis $\{m_1, \dots, m_n \}$of a Hilbert space $H$ (or just think it as $\mathbb R^n$, and I think the methods should be the same) with given inner product, how can we find the coordinates of a vector $v$ in the space? Is there a solution in some elegant form for the coordinates?

I am thinking about first apply Gram-schmidt method to orthogonalize the basis, but feel the derivation too complicated.

Thanks.

Answer 1

The vector of coordinates $x$ can be easily found by a linear system $Ax = b$ where $A$ is the Gram Matrix of the base ${\{m_1,...,m_n\}}$ and $b$ is the vector $\{<v,m_i>\}$

Answer 2

It depends on how $\{m_1,...,m_n\}, v$ are "given". If they are given as coordinate column-vectors in some other basis, then the solution to the matrix equation $Ax=v$ where $A = [m_1 \mid m_2 \mid \dots \mid m_n]$ gives the coordinates in the new basis. If their coordinates are unknown, but, for example, we can compute their inner products, then the coordinates can be found from $A^TAx=A^Tv$ as in Tal's answer.