It seems like I'm a bit confused about the transition matrices , my question are as follows:
$1.$ If $T$ is a transition matrix from a basis $B$ to the standard basis , do we have $T.v_B=v_S?$ where $v_B$ is the vector $v$ expressed in the basis $B$ and $v_S$ is the vector $v$ expressed in the standard basis.
$2.$ If $P_B$ is the matrix of a linear transformation with respect to $B$ , Is the matrix of the same linear transformation with respect to the standard basis equal to : $T.P_B.T^{-1}$?
Thanks in advance!