Answer: The required matrix is
[tex]T=\left[\begin{array}{ccc}-1&3\\2&4\end{array}\right] .[/tex]
Step-by-step explanation: We are given to find the transition matrix from the bases B to B' as given below :
B = {(-1,2), (3, 4)) and B' = {(1, 0), (0, 1)}.
Let us consider two real numbers a, b such that
[tex](-1,2)=a(1,0)+b(0,1)\\\\\Rightarrow (-1,2)=(a,b)\\\\\Rightarrow a=-1,b=2.[/tex]
Again, let us consider reals c and d such that
[tex](3,4)=c(1,0)+d(0,1)\\\\\Rightarrow (3,4)=(c,d)\\\\\Rightarrow c=3,d=4.[/tex]
Therefore, the transition matrix is given by
[tex]T=\left[\begin{array}{ccc}-1&3\\2&4\end{array}\right] .[/tex]
Thus, the required matrix is
[tex]T=\left[\begin{array}{ccc}-1&3\\2&4\end{array}\right] .[/tex]
