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jclawson709
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Homework Statement
Let V = P_n(F), and let c_0, c_1,..., c_n be distinct scalars in F. For 0 <= i <= n, define f_i(p(x)) = p(c_i). Prove that {f_0, f_1,..., f_n} is a basis for V*. Hint: Apply any linear combination of this set that equals the zero transformation to p(x) = (x-c_1)*(x-c_2)*...*(x-c_n), and deduce that the first coefficient is zero.
Homework Equations
The Attempt at a Solution
I really have no clue where to start on this one. First off, I don't even see how doing what the hint says would prove it to be a basis. Is the hint implying that these scalars are the roots of the polynomial? Also I'm not sure how I would pick a linear combination of the set that equals 0. I was thinking of using c_i - a*c_j, where a is just another scalar which causes a*c_j = c_i, so that it equals zero and the resulting transformation equals zero. Not too sure about that reasoning though, and how it would play into the hint the book gives.
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