Change of basis matrix linear transformation

Question

Let $\pmatrix{2&-1\\-3&2}$ be the change of basis matrix from the basis V to the basis W $(P_{wv})$ in the subspace U.

Let $f: U -> U$ be a linear transformation represented by the matrix $ [f]_w\pmatrix{2&1\\2&-1}$. Decide $[f]_v$:

Okay, so first of all I found $P_{vw}$ which is: $\pmatrix{2&1\\3&2}$. To transform $[f]_w$ to $[f]_v$ wouldn't I just do $[P]_{vw}[f]_w$? Shouldn't that change $[f]_w$ with respect to the basis v?

Answer 1

Let $M=\pmatrix{2&-1\\-3&2}$, note that

$$u_w=Mu_v\implies u_v=M^{-1}u_w$$

then

$$y_w= [f]_wx_w\implies My_v= [f]_wMx_v \implies y_v= M^{-1}[f]_wMx_v$$

and therefore $$[f]_v=M^{-1}[f]_wM$$