Chapter 3.4, Problem 23E

In Exercises 19 through 24, find the matrix B of the linear transformation $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ with respect to the basis $=\left({\stackrel{\to }{v}}_1,{\stackrel{\to }{v}}_2\right)$. For practice, solve each problem in three ways: (a) Use the formula $B=S^{-1}AS$, (b) use a commutative diagram (as in Examples 3 and 4), and (c) construct B “column by column.”

23. $A=\left[\begin{array}{cc}5& -3\\ 6& -4\end{array}\right];{\stackrel{\to }{v}}_1=\left[\begin{array}{l}1\\ 1\end{array}\right],{\stackrel{\to }{v}}_2=\left[\begin{array}{l}1\\ 2\end{array}\right]$