"The matrix representation of $T:R^2 -> R^2$ in the basis {$\begin{pmatrix} 1 \\ 1 \\ \end{pmatrix}$, $\begin{pmatrix} -2 \\ 2 \\ \end{pmatrix}$} is $D =\begin{pmatrix} 1 & 0 \\ 0 & \frac 1 {2} \\ \end{pmatrix}$
Decide $T^n ( \begin{pmatrix} 2 \\ 5 \\ \end{pmatrix})$ for every integer $n>0$."
Isn't $T(x)$ representation of the linear transformation which means that I have: $\begin{pmatrix} 1 & 0 \\ 0 & \frac 1 {2} \\ \end{pmatrix}^n \begin{pmatrix} 2 \\ 5 \\ \end{pmatrix}$? Or how do I tackle this problem?
Another guess that I have is that I should use that $ A^n = PD^nP^{-1}$ but I don't know if that's applicable here?
It is incorrect to say that $$ T^n \pmatrix{2\\5} = \begin{pmatrix} 1 & 0 \\ 0 & \frac 1 {2} \\ \end{pmatrix}^n \begin{pmatrix} 2 \\ 5 \\ \end{pmatrix} $$ In particular, note that the diagonal matrix is the matrix with respect to the basis given. Instead, note that $$ \pmatrix{2\\5} = 3.5 \pmatrix{1\\1} + .75 \pmatrix{-2\\2} $$ and proceed from there.