Question

Let F be a field (for instance R or C), and let P2(F) be the set...

Let F be a field (for instance R or C), and let P2(F) be the set of polynomials of degree ≤ 2 with coefficients in F, i.e.,

P2(F) = {a0 + a1x + a2x2 | a0,a1,a2 ∈ F}.

Prove that P2(F) is a vector space over F with sum ⊕ and scalar multiplication defined as follows:

(a0 + a1x + a2x^2)⊕(b0 + b1x + b2x^2) = (a0 + b0) + (a1 + b1)x + (a2 + b2)x^2

λ (b0 + b1x + b2x^2) = (λ·b0) + (λ·b1)x + (λ·b2)x^2

How about the set of polynomials of exactly degree 2 with coefficients in F, is it a vector space over F?

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Answer #1

Let F be a field and let be the set of polynomials of degree with coefficients in F.

Let -- and .

Hence P2(F) is a vector space over F with sum .

Hence P2(F) is a vector space with scalar multiplication.

Now Let consider the set of the polynomials of exactly degree 2.

This is not a vector space over F. We can understand it by a counterexample.

Let -- and .

Since the is now the set of polynomials of exactly degree 2. So 0 not belong to .

So, not belong to P2(F).

Hence it is not a vector space.

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