Question

consider the operation on p2 that takes ax^2+bx+c to cx^2+bx+a. does it correspond to linear transformation from R^3to R^3 .if so what is its matrix

Answer 1

Yes it does (checking that would mean checking that it satisfies the definition). To find the matrix, we want to know where the basis elements map to. Recall that ax^2+bx+c corresponds to [a,b,c] as a vector, and has the usual basis there. (Read these as column vectors).

Where does [1,0,0] map to? Well, ax^2 goes to a, so [0,0,1].
Where does [0,1,0] map to? bx goes to bx, so [0,1,0].
Where does [0,0,1] map to? c goes to cx^2, so [1,0,0].

And these vectors give you the matrix:
\[ \pmatrix{0 & 0 & 1\\ 0 & 1 & 0 \\ 1 & 0 & 0}\]