# Finding T*x given T and x

#### 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 $$\displaystyle \left[ \begin{array}{c} 11 \\ -12 \end{array} \right]$$ .
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*

#### tonio

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*
.

#### kaelbu

I used the formula $$\displaystyle T*_{ij} = (-1)^{i+j} det [j ,i]$$ , I think ... on second thought (this actually gives $$\displaystyle \left[ \begin{array}{cc} 2 & -1 \\ -1 & -3 \end{array} \right]$$, I think ) that may have not been what I used, but it's what I MEANT to use. But that really is beside the point, because I most certainly do not know how to use this mysterious formula for my next problem.
I tried to use the orthonormal basis $$\displaystyle \beta = { e_1 , e_2 }$$ And do something involving $$\displaystyle T(x) = < x, y>$$ where $$\displaystyle y= T(e_1) e_1 + T(e_2)e_2$$ which = -1. But I had no idea where to go from there or if that was anywhere near what I supposed to do.