I am very confused about this question can someone please help me solve it. Thanks!

Question

Lucy works in the space $\mathbb{R}^n$ of vectors $x = [x_1, . . . , x_n]^T$ . Chris chosen a different basis and handles vectors $x’ = [x’_1 , . . . , x’_n ]^T$ , where $x$ and $x’$ are related by $x = Px’$ . Lucy is studying the linear transformation in $\mathbb{R}^n $ associated with the $n\times n$ real matrix $A$, i.e. $x \mapsto y = Ax$. How shall we find the coordinates y’ of the transformed vector when we know the coordinates $x’$?

Answer 1

$y’=Py$ and $x’=Px$. Moreover $y=Ax$ so that

$y’=Py=PAx=PAP^{-1}x’$

Thus $y’=PAP^{-1}x’$