Change of basis and independence problem

Question

I'm asking about b), what does "independent of the choice of the matrix" mean exactly? What do we need to show? Can somebody give a hint please...

Answer 1

$A = \begin{bmatrix} 3&5\\1&2 \end{bmatrix}$ will take vector $x_{a'}$ (in terms of the basis $a'$) to is corresponding vector in term of the basis $a.$

and $A^{-1}$ will take a vector in terms of the basis $a$ to its eqivalent in basis $a'.$

$A^{-1}TA\mathbf x_{a'}$ will take a vector in the basis $a'$ take it to the basis $a$ transform it in the basis $a$ and take the result back to the basis $a'$