So I have the question Let F be a field and let L be the set of all polynomials f(x) element of F[X] satisfying the condition that deg(f) is even. Is L a subspace of F[X]?

I would say that L is not a subspace of F[X].

Taking the condition f(-a)=-f(a)

if we took f(x)=x^2 -2, which has an even degree, then

f(-a)=(-a)^2 -2 = a^2 -2 which does not equal -(x^2 -2)

Is my approach correct?

  • $\begingroup$ Where did the condition $f(-a)=-f(a)$ come from? That's what it means for the polynomial $f$ to be an odd function, but actually being an even function ($f(-x)=f(x)$) isn't the same as having even degree. $\endgroup$ – Nate Eldredge Jun 18 '14 at 18:31
  • $\begingroup$ You need to distinguish between the concepts of even degree and even as a function; the former means the leading coefficient of $f(x)$ is divisible by $2$; that latter means $f(-a) = f(a)$ for any $a \in F$. $\endgroup$ – Robert Lewis Jun 18 '14 at 18:32
  • $\begingroup$ @NateEldredge: whoops! Corrected! Thanks for pointing that out! $\endgroup$ – Robert Lewis Jun 18 '14 at 18:34
  • $\begingroup$ @RobertLewis, thank you, yes I understand the difference now, silly mistake. $\endgroup$ – cele Jun 18 '14 at 19:07

Even degree polynomial is NOT the same as even function. For example: $x^2+x$ is an even degree polynomial but not an even function. So if your set $L$ has even degree polynomials then it is NOT a subspace because $f(x)=x^2+x$ and $g(x)=-x^2$ are both in $L$ but $f+g \not\in L$.


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