Originally Posted by

**kaelbu** The problem: V= $\displaystyle R^2$ , T(a,b)= (2a+b, a-3b), x= (3,5). Evaluate T* at the given vector in V.

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] $ .

Uh?? And how do you *actually* find what $\displaystyle T^{*}$ is?? __This__ is the problem in this question!

Yet if we remember that $\displaystyle T^{*}$ is easily representable by means of $\displaystyle T$ when we choose an orthonormal basis for our vector space then the problem is easy.

Tonio

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*