Homework problem
Consider the basis {1, x,(x − 1)^2} for P2 and let F : P2 → P2 be a linear transformation.
Let F(1) = (x + 1)^2, F(x) = x, F((x-1))^2.
Find the rule for F(a x^2 + b x + c).
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You should at least make some efforts to make your question intelligible!
This looks like homework. If you want some help, you should ask more precise questions related to your research in solving those exercises, especially where you are locked.
More precisely, this question doesn't look like it it related to Sage!
I guess P2 is the vector space of polynomials of degree less or equal two. Then we get a map F on the system 1, x, (x-1)^2 with values in the same space, which we extend linearly. Unfortunately the map is defined only on the first two elements, for the third one we run into human compiler error.