Finding a Basis for a Subspace with a Specific Condition?

[phy]

Let V be the subspace of F([0,1],R) generated by the functions f1, f2, f3 given by:

f1(x)=1/(x+1) , f2 (x) = 2-x and f3(x) = x^2

for all x element of [0,1]. Find a basis of the subspace U of V that consists of all the functions g of V such that g(0) = g(1).

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Ok now the very first question I have is what on Earth is g? We have three functions (f1, f2, and f3) so where did this g come from? Secondly, how do I start this question? I've looked in my textbook and lecture notes but there aren't any examples like this one. We only have the really simple vector subspace examples and they didn't really help much. Any suggestions would be greatly appreciated. Thanks

 

[phy]

Does nobody know how to do this question at all?

 

[Fredrik]

Huh? You defined what g is, so why are you asking?

Any g in U must satisfy

g(0)=g(1)

and since U is a subset of V, g must also satisfy

g(x)=a_1f_1(x)+a_2f_2(x)+a_3f_3(x)

where the a's are real numbers. What you will have to figure out is how the condition g(0)=g(1) restricts the possible values of a1, a2 and a3.