I have tried to solve the following:- Let $P_2 -> P_2$ be defined by $$T(a+bx+cx^2) = 2b+3cx+(a-b)x^2$$ Find the matrix of transformation w.r.t ordered bases $B_1= \left\{x^2, x^2+x, x^2+x+1\right\}$ and $B_2 = \left\{1,x,x^2 \right\}$
What I have done :- Considering $(1,0,0) (0,1,0)\;\; \text{and}\;\; (0,0,1)$ as the standard ordered basis So, When $T(1+0x+0x^2) 2b+3cx+(a-b)x^2=x^2$
$T(0+1x+0x^2) 2b+3cx+(a-b)x^2=2-x^2$
$T(0+0x+1x^2)\;\; \text{gives}\;\; 3x$ Therefore the matrix is:- 0 2 0 0 0 3 1 -1 0 Now the transition matrices are :- B1 0 1 1 0 0 1 1 1 1 B2 1 0 0 0 1 0 0 0 1 As the formula Pe^-1 <-B2 At Pe<-B1 Inverse of B2 x Matrix from polynomial x B1 gave me the following result:- 0 0 2 3 3 3 0 0 -1 Is it correct?