Finding T*x given T and x

May 2010
13
3
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*
 
Oct 2009
4,261
1,836
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*
.
 
May 2010
13
3
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.