Chapter 4.3, Problem 33E

In Exercises 5 through 40, find the matrix of the given linear transformation T with respect to the given basis. If no basis is specified, use the standard basis: $\mathfrak{A}=\left(1,t,t^2\right)$for $P_2$,

$\mathfrak{A}=\left(\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right)$

for ${ℝ}^{2\times 2}$, and $\mathfrak{A}=\left(1,i\right)$for $ℂ$. For the space $U^{2\times 2}$of upper triangular $2\times 2$matrices, use the basis

$\mathfrak{A}=\left(\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right)$

unless another basis is given. In each case, determine whether T is an isomorphism. If T isn‘t an isomorphism, find bases of the kernel and image of T, and thus determine the rank of T.

33. $T\left(f\left(t\right)\right)=f\left(1\right)+f^{\prime }\left(1\right)\left(t-1\right)$ from $P_2$ to $P_2$,with respect to the basis $=\left(1,t-1,{\left(t-1\right)}^2\right)$