Question

If e is a projection and f is a polynomial, then fee) = ai + be. What are a and b in terms of the coefficients of f?

Answer 1

Approach: Let f(x)=a0+a1x+...+anxn, where ai∈F.

Now, f(E)=a0I+a1E+...+anEn. But, the projection map being idempotent by definition, E=E2=...=En. Hence, we end up with f(E)=a0I+(a1+...+an)E.

Thereby, a=a0, and b=a1+...+an.

Is this okay? The wording of the question is a bit too confusing for me to be sure of what I am doing here, so a verification would be nice.

Thank you.