Let matrix $T = \left(\begin{array}{cc}1 & 2 \\1 & -1\end{array}\right).$ Let $e_1 = (1,1)^T$ and $e_2 = (3,1)^T$. $T$ is currently relative to the standard basis. If asked to find $T$ relative to bases $e_1,e_2$, do you solve the problem via:
(a) $H$ \ $T$ (note: matlab notation; this solves for $x$ in $Hx = T$).
or (b) $H^{-1} * T * H$?
where $H$ = $\left(\begin{array}{cc}1 & 3 \\-1 & 1\end{array}\right).$
I'm confused because question: Find matrix of linear transformation seems to suggest that (b) is the way to go, but the question Find matrix of linear transformation relative to new bases seems to suggest that a way of representing a matrix relative to a new basis is more like (a) [not exactly the same since $T$ is not invertible, but since $T$ for us is invertible, we can ignore this fact.]