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Then, a sample element of ##S## would look like:

$$

p(t) = c_0 + c_1t +c_2t^2 + \cdots + c_nt^n

$$

Now, to satisfy ##p(0)=p(1)## we must have

$$

\sum_{i=1}^{n} c_i =0

$$

What could be the possible bases for S? I thought of one of them and it looks like this

$$

A = \{ 1, c_1t +c_2t^2, c_1t +c_2t^2+c_3t^3, \cdots c_1t +c_2t^2 ... +c_nt^n | \sum_{i=1}^{j} c_i =0 ; j=1,2,3 ... n\}

$$

Is A a basis for S? I mean, I'm unable to disprove it.