<s>[INST] Linear transform of polynomials The following question is from Golan's linear algebra book. I have posted a solution in the answers. 
Problem: Let $F$ ba field and let $V$ be a vector subspace of $F[x]$ consisting of all polynomials of degree at most 2. Let $\alpha:V\rightarrow F[x]$ be a linear transformation satisfying
$\alpha(1)=x$
$\alpha(x+1)=x^5+x^3$
$\alpha(x^2+x+1)=x^4-x^2+1$.
Determine $\alpha(x^2-x)$.
 [/INST] The idea here is to determine the action on each of the basis elements $1$, $x$, and $x^2$.. By linearity we see 
$\alpha(x)=\alpha((x+1)-1)=\alpha(x+1)-\alpha(1)=x^5+x^3-x$
$\alpha(x^2)=\alpha((x^2+x+1)-(x+1))=\alpha(x^2+x+1)-\alpha(x+1)=-x^5+x^4-x^3-x^2+1$
Using linearity again we see
$\alpha(x^2-x)=\alpha(x^2)-\alpha(x)=-x^5+x^4-x^3-x^2+1-x^5-x^3+x=-2x^5+x^4-2x^3-x^2+x+1$
and if the field is of characteristic 2, the $x^5$ and $x^3$ terms disappear. 
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