Answer
See below
Work Step by Step
a) Obtain:
$T(1)=x+1\\
T(x)=x^2+x\\
T(x^2)=x^3+x^2$
then $T(1)=x+1=1.1+1.x+0.x^2+0.x^3\\
T(x)=x^2+x=0.1+1.x+1.x^2+0.x^3\\
T(x^2)=x^3+x^2=0.1+0.x+1.x^2+1.x^3$
We have:
$[T(1)]_C=\begin{bmatrix}
1\\1 \\ 0\\ 0
\end{bmatrix}\\
[T(x)]_C=\begin{bmatrix}
0\\
1\\1\\0
\end{bmatrix}\\
[T(x^2)]_C=\begin{bmatrix}
0\\
0 \\ 1 \\ 1
\end{bmatrix}$
Hence, $[T]^C_B=\begin{bmatrix}
1 & 0 & 0\\1 & 1 & 0\\ 0 &1 & 1\\0 & 0 & 1
\end{bmatrix}$
b) Obtain:
$T(1)=x+1\\
T(x-1)=(x+1)(x-1)\\
T[(x+1)^2]=(x+1)(x-1)^2$
then $T(1)=x+1=2.1+1.(x-1)+0.(x-1)^2+0.(x-1)^3\\
T(x-1)=(x+1)(x-1)=(x-1+2)(x-1)=(x-1)^2+2.(x-1)=0.1+2.(x-1)+1.(x-1)^2+0.(x-1)^3\\
T[(x-1)^2]=(x+1)(x-1)^2=(x-1+2)(x-1)^2=(x-1)^3+2.(x-1)^2=0.1+0.(x-1)+2.(x-1)^2+1.(x-1)^3$
We have:
$[T(1)]_C=\begin{bmatrix}
2\\1 \\ 0\\ 0
\end{bmatrix}\\
[T(x)]_C=\begin{bmatrix}
0\\
2\\1\\0
\end{bmatrix}\\
[T(x^2)]_C=\begin{bmatrix}
0\\
0 \\ 2 \\ 1
\end{bmatrix}$
Hence, $[T]^C_B=\begin{bmatrix}
2& 0 & 0\\1 & 2 & 0\\ 0 &1 & 2\\0 & 0 & 1
\end{bmatrix}
