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jaejoon89
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Homework Statement
Prove or disprove: there exists a basis (p_0, p_1, p_2, p_3) of P_3 (F) such that one of the polynomials p_0, p_1, p_2, p_3 has degree 2.
Homework Equations
none really
The Attempt at a Solution
Is the following proof correct?
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Let p_0, p_1, p_2, p_3 be elements of P_3(F) s.t.
p_o (x) = 1,
p_1 (x) = x,
p_2 (x) = x^2 + x^3,
p_3(x) = x^3.
None of the polynomials are degree 2 although (p_0,p_1,p_2,p_3) is clearly spanning P_3 (F) with dimP_3(F) = 4 and forms a basis. Hence proved.