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.
Best Answer
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
$$\begin{pmatrix}2&\!\!-1\\5&\!\!-3\end{pmatrix}$$