Good Day!
I pretty much have a rough idea on how to deal with matrices and vectors but what boggles me are items that are in polynomials. I can basically start answering but can't seem to finish it for example:
$$L:P_2\rightarrow P_1\ is\ given\ by\ L(p(t))=p'(t)$$ having the ordered bases of $$ S=[t^2,t,1] \ and \ G=[t,1] $$ Determine the matrix of L with respect to S to G.
From here I know that I should set $$p(t)=at^2+bt+c$$ and then getting its derivative $$p'(t)=2at+b$$ then I know that I should substitute S to L(t): $$ L(t^2)=2t$$$$L(t)=1$$ $$L(1)=0$$
So from this point I am quite lost I have no idea on how I should proceed even the substituting part seems hazy to me I understood that you differentiate the value itself (the one in S) but I don't understand why just I don't substitute the value to p'(t). The book I have did not really explain why that happened. Any tips or hints on how I could go about this I am not really looking for the answer, I am just looking for the process on how I could solve this. Thank you very much in advance!