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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.

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I don't understand the question. Unless there's more structure on the vector space — e.g. as you say an inner product, or a pre-existing basis, I don't think this can be done explicitly (but it can be done, by definition). – Zhen Lin Jan 5 '11 at 10:12
It depends on the particular vector space and how you are "given" the basis vectors $e_i$; but most of the time, you can often set up the problem as a system of linear equations that can be solved by the usual way (e.g., Gauss-Jordan elimination). – Arturo Magidin Jan 5 '11 at 14:30
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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)

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You could use the Gram-Schmidt process to create an orthonormal basis.

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If you have another basis in which you can express $v$ and $e_i$, finding $\lambda_i$ reduces to solving system of linear equations.

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