Prove that if V = Pn(R) and S = {p1, p2,..., pk } is a set of vectors in V each of a
Chapter 4, Problem 54(choose chapter or problem)
Prove that if V = Pn(R) and S = {p1, p2,..., pk } is a set of vectors in V each of a different degree, then S is linearly independent. [Hint: Assume without loss of generality that the polynomials are ordered in descending degree: deg(p1) > deg(p2) > > deg(pk ). Assuming that c1 p1 + c2 p2 ++ ck pk = 0, first show that c1 is zero by examining the highest degree. Then repeat for lower degrees to show successively that c2 = 0, c3 = 0, and so on.]
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