Finding Linear Transformation Matrix 
Find the matrix $A$ of a linear transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ that satisfies $$T\left[2\choose3\right] = {1\choose1}, \ T^2\left[{2\choose3} \right]= {1\choose2}.$$

I am trying to review some linear algebra and was confused about this question.
The answer given is $$A = \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix}$$ and I am not sure how it was obtained.
 A: Since it must be
$$T^2\binom23=T\left(T\binom23\right)=T\binom11=\binom12$$
we get, wrt the standard basis (BTW, check $\;\left\{\;\binom23\;,\;\;\binom11\;\right\}\;$ (check this is actually a basis!), first that
$$\begin{align}\binom10&=(-1)\binom23+3\binom11\\{}\\
\binom01&=1\cdot\binom23+(-2)\binom11\end{align}$$
we get that
$$\begin{align}&T\binom10:=T\left(-1\binom23+3\binom11\right)=-1\binom11+3\binom12=\binom25=2\binom10+5\binom01\\{}\\
&T\binom01=T\left(1\binom23-2\binom11\right)=\binom11-2\binom12=\binom{-1}{-3}=(-1)\binom10+(-3)\binom01\end{align}$$
and I thus get the matrix
$$\begin{pmatrix}2&\!\!-1\\5&\!\!-3\end{pmatrix}$$
A: The linear transformation $T$ is defined by
$$T((2,3)^T)=(1,1)^T\quad\text{and}\quad T((1,1)^T)=(1,2)^T$$
since $B=((2,3)^T,(1,1)^T)$ is a basis for $\Bbb R^2$. Let $P$ the matrix change from the standard basis to $B$ then
$$P=\begin{pmatrix}2&1\\3&1\end{pmatrix}$$
so the matrix of $T$ relative to the standard basis is given by
$$[T]=\begin{pmatrix}1&1\\1&2\end{pmatrix}P^{-1}=\begin{pmatrix}2&\!\!-1\\5&\!\!-3\end{pmatrix}$$
A: T^2[(2,3)] = (1,2) implies that T(T(2,3)) = (1,2) i.e. T(1,1) = (1,2).
The general linear transformation is not defined here. However, two linearly independent vectors are given. So, any general say, (x,y) vector can be given by the linear combination of these two vectors.
(x,y) = a*(2,3) + b*(1,1) -- (I)
On comparing we will get two equations in two unknowns x = 2a + b and y = 3a + b.
From these two equations we can find the values of a and b in terms of x,y. Now substitute these in (I). Thus, we get the general co-ordinate and hence the general linear transformation as,
T(x,y) = aT(2,3) +bT(1,1) = (-x-y)(1,1) + (3x-2y)(1,2).
Now we can find the matrix of linear transformation with respect to the standard basis of R^2.
Hope this helps!
