Hi all!

I got a problem, I need to make a change of basis, I have a matrix representation

$\displaystyle T= {ab }\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 &0 \\ 0 & 0 & 3 \end{bmatrix} $ in the basis$\displaystyle B= \left\{{ \begin{bmatrix} 1\\ 0 \\ \ 0\end{bmatrix} , \begin{bmatrix} 0 \\ 1 \\ \ 0\end{bmatrix} , \begin{bmatrix} 0\\ 0 \\ \ 1\end{bmatrix} }\right\} = e_j$

and now I have to put the matrix T in the basis

$\displaystyle B '= \left\{{ \begin{bmatrix} -1\\ 1 \\ \ 0\end{bmatrix} , \begin{bmatrix} 1\\ 1 \\ \ 0\end{bmatrix} , \begin{bmatrix} 0\\ 0 \\ \ 1\end{bmatrix} }\right\} = e' _{i}$

I have tried ...

The matrix change of basis from B to B' is

$\displaystyle A =a_{ij}$ where $\displaystyle a_{ij} = < e'_{i} , e_{j} >$

The inners products are

$\displaystyle < e'_{1} , e_{1} > = [ -1 1 0 ] \begin{bmatrix} 1\\ 0 \\ \ 0\end{bmatrix}= -1$

and so on..

therefore the matrix A is

$\displaystyle A= \begin{bmatrix} -1 & 1 & 0 \\ 1 & 1 &0 \\ 0 & 0 & 1 \end{bmatrix}$ , finally the matrix T in the basis B' is equal to $\displaystyle A T A ^T$

$\displaystyle A^T$ is the matrix transpose, It is correct or wrong ?

Thanks in advance!