Thoughts:

I solved this by putting T in matrix form ( \(\displaystyle \left[ \begin{array}{cc} 2 & 1 \\ 1 &-3 \end{array} \right] \) ) finding \(\displaystyle T^*\) ( \(\displaystyle \left[ \begin{array}{cc} 2 & 1 \\ 1 &-3 \end{array} \right] \) ) and computing \(\displaystyle T^*x\) which equals \(\displaystyle \left[ \begin{array}{c} 11 \\ -12 \end{array} \right] \) .

I believe this method to be correct (though I wouldn't really be shocked if you told me it wasn't), however, I'd like to be able to solve this problem using the property \(\displaystyle <T(X), y> = <x, T^*(y)> \) because it is my understanding that this is the only way to solve some of these problems. But I can't figure out how this is possible despite at least an hours attempt. *bangs head against wall*