Chapter 4.7, Problem 56E

True or False? In Exercises $55$and $56$, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows that the statement is not true in all cases or cite an appropriate statement from the text.

(a) If $P$ is the transition matrix from a basis $B^{\prime }$ to $B$, then $P^{-1}$ is the transition matrix from $B$ to $B^{\prime }$.

(b) To perform the change of basis from a nonstandard basis $B^{\prime }$ to the standard basis $B$, the transition matrix $P^{-1}$ is simply $B^{\prime }$.

(c) The coordinate matrix of $p=-3+x+5x^2$ relative to the standard basis for $P_2$ is ${\left[p\right]}_S={\left[\begin{array}{ccc}5& 1& -3\end{array}\right]}^T$.