Prove/Disprove: Existence of a Basis of P_3(F) w/ Degree 2 Polynomial

[jaejoon89]

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.

 

[LCKurtz]

Looks good to me assuming the "one" in the statement of the problem should be "none".

 

[aPhilosopher]

I'm not sure that stating that it clearly spans it will suffice even if it is obvious. If you think that this suffices for you class, you're fine.

On the other hand, you could cook up a matrix that maps a degree three polynomial represented in the standard basis to it's representation in this basis pretty easily.

 

[jaejoon89]

Typo: the original statement is supposed to be "none of the polynomials has degree 2." Thanks for pointing that out, LCKurtz.