Chapter 5.10, Problem 14E

In Exercises 14-16, proceed through the following steps:

a. Find the matrix, $Q$, of $T$ with respect to the natural basis $B$ for $V$.

b. Show that $Q$ is similar to a diagonal matrix; that is, find a nonsingular matrix $S$ and a diagonal matrix $R$ such that $R=S^{-1}QS$.

c. Exhibit a basis $C$ of $V$ such that $R$ is the matrix representation of $T$ with respect to $C$.

d. Calculate the transition matrix, $P$ from $B$ to $C$.

e. Use the transition matrix $P$ and the formula $R{\left[v\right]}_C={\left[T\left(v\right)\right]}_C$ to calculate $T\left(w_1\right)$, $T\left(w_2\right)$, and $T\left(w_3\right)$.

$V={\mathcal{P}}_1$ and $T:V\to V$ is defined by $T\left(a_0+a_{1}x\right)=\left(4a_0+3a_1\right)+\left(-2a_0-3a_1\right)x$. Also,

$w_1=2+3x,w_2=-1+x,$ and $w_3=x$.