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 $\alpha(x^2-x)$.
Answer
Your calculation looks fine. Equivalently, but with rather less work, note that
$$x^2-x=(x^2+x+1)-2(x+1)+1,$$
so we can compute $\alpha(x^2-x)$ directly by linearity, without computing $\alpha$ at the most common basis elements.
No comments:
Post a Comment