The transition matrix from $\alpha$ to $\alpha'$ is composed of the coordinates in $\alpha$ of vectors from $\alpha'$ written as columns. I.e., if $\alpha'=(v_1',v_2',v_3')$, we write the coordinates of $v_1'$ in $\alpha$ as the first column, the coordinates of $v_2'$ in $\alpha$ as the second column and so on.Note: some people call this the transition matrixfrom$\alpha'$to$\alpha$. Don't have anything to do with them. Kidding.

Let $C$ be the transition matrix from $\alpha$ to $\alpha'$. If $v=(x,y,z)$ in $\alpha$ and $v=(x',y',z')$ in $\alpha'$, then

\[

\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}= C\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}

\]

("old" coordinates are expressed through "new" ones). This makes sense: $v$ is expressed through $v_1',v_2',v_3'$ using coefficients $x_1',x_2',x_3'$, and each of $v_i'$'s is expressed through $v_j$'s using the column vectors of $C$. Ultimately, $v$ is expressed though $v_j$'s. Note that since $C$ converts "primed" coordinates $x_1',x_2',x_3'$ into "non-primed" coordinates $x_1,x_2,x_3$, some people call $C$ the transition matrix from $\alpha'$ to $\alpha$.

If $C$ is the transition matrix from $\alpha$ to $\alpha'$ in my sense and $A_T$ is the matrix of $T$ in $\alpha$, then the matrix of $T$ in $\alpha'$ is $C^{-1}A_TC$. Indeed, the matrix of an operator converts coordinates of a vector in some basis into the coordinates of theimageof that vector in the same basis ("new" coordinates are expressed through "old" ones; compare this with the transition matrix). So if $v=(x_1,x_2,x_3)$ in $\alpha$, then $Tv$ has coordinates $A_T\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}$ in $\alpha$. Meanwhile, let $v=(x_1',x_2',x_3')$ in $\alpha'$. Then $v$ has coordinates $C\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}$ in $\alpha$, $Tv$ has coordinates $A_TC\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix}$ in $\alpha$, and finally $Tv$ has coordinates $C^{-1}A_TC\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix} $ in $\alpha'$.

So, to solve your problem:

(1) write the transition matrix $C$ from $\alpha$ to $\alpha'$,

(2) compute $C^{-1}A_TC$.