(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let S be any non-empty set, F be a field and V={ f : S -> F such that f(x) = 0 } be a vector space over F.

Let f[sub k] (x) : S -> F such that f[sub k] (x) = 1 for k=x, otherwise f[sub k] (x) = 0.

Prove that the set { f [sub k] } with k from S is a basis for the vector space V.

3. The attempt at a solution

I tried to sketch something but i am not sure i'm on the right path.

So, given B={ f [sub k] }, k from S, it is a basis for V if and only if B spans any vector from V and B is linearly independent.

Let g : S -> F be a vector from V, then g(x)=0 and asome scalars from F with i >= 1.

Then B spans g if and only if g = sum ( a* f).

But g(x) = 0 so 0 = sum ( a* f), so the vectors fare linearly independent.

So i'd say B is a basis for the vector V, but i'm not sure it's correct because i didn't make use of the definition of the function f.

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# Homework Help: Vector space basis proof

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