Chapter 5.5, Problem 23E

By Theorem 5 of Section 5.4, an $\left(n\times n\right)$ transition matrix (see Exercise 21 and 22) is always nonsingular. Thus if ${\left[w\right]}_B=A{\left[w\right]}_C$, then ${\left[w\right]}_C=A^{-1}{\left[w\right]}_B$. Calculate $A^{-1}$ for the matrix in part a) of Exercise 21 and use the result to transform each of the following polynomials to the form $a_0+a_{1}x+a_2{x}^2$.

a) $p\left(x\right)=2-3\left(x+1\right)+7{\left(x+1\right)}^2$

b) $p\left(x\right)=1+4\left(x+1\right)-{\left(x+1\right)}^2$

c) $p\left(x\right)=4+\left(x+1\right)$

d) $p\left(x\right)=9-{\left(x+1\right)}^2$