This subject came up in some notes on linear algebra I'm reading and I don't get it. Please help me understand.(adsbygoogle = window.adsbygoogle || []).push({});

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First, the relevant background and notation relating to my question:

Let S be a nonempty set and F be a field. Denote by l(S) the family of all F-valued functions on S and l_c(S) the family of all functions mapping S to F with finite support; that is, those functions on S which are nonzero at only finitely many elements of S.

Now let V be a vector space with (over F) with basis B. Define M : l_{c}(B) \rightarrow V by M(v) = sum_{e \in B}v(e)e.

Note that l(B) (under pointwise operations) is a vector space, l_c(B) is a vector subspace of l(B), and M is an isomorphism.

If we writevfor M(v) we see thatv= sum_{e \in B}v(e)e.

We will go further and use M to identify V with l_c(B) and write v = sum_{e \in B}v(e)e. That is, in a vector space with basis we will treat a vector as a scalar valued function on its basis.

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And now here is the question: according to the above, what is the value of f(e) when e and f are elements of the basis B? (And most importantly,why?)

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