I am given a vector space $V$, a basis $e_1,\dots,e_n$ and a vector $v\in V$ and I am asked to find $\lambda_1,\dots\lambda_n$ from the underlying field such that $v=\sum \lambda_i e_i$.
How can I do it?
Of course, if I know the basis is orthonormal then I can compute the inner product between $v$ and the $e_i$'s, but let's assume I don't know this, so the question is whether I can solve this problem in this level of abstraction.
Best Answer
In the absence of an inner product you could use any n independent linear functionals to get n equations in n unknowns. (eg for a function space of solutions to a Boundary Value Problem you might just use evaluation at n points in the domain)