The problem is prove that $\beta$={$f_0(x),f_1(x)...f_n(x)$} is basis for $P_n(R)$
($P_n(R)$={$a_0+a_1x+..+a_nx^n|$ $a_i\in$R} )
$$f_i(x)=\frac{(x-c_0)...(x-c_{i-1})(x-c_{i+1})...(x-c_n)}{(c_i-c_0)...(c_i-c_{i-1})(c_i-c_{i+1})...(c_i-c_n)}$$ ($c_0$, $c_1$...,$c_n$ are distinct elements in R)
Please help me. Thank you.
It is enough to show that they are independent (make sure you understand why).
So suppose $a_0f_0 + a_1 f_1 + a_2f_2 + \dots + a_n f_n = 0$ and start plugging in values for $x$ to show that the $a$'s are all 0.