This tag is for question about changing basis of a finite dimensional vector space. For example, how does the representation of a vector, or a matrix change with the change of basis. Please don't use this tag on its own, it is better to add a more general tag which is relevant to your question, e.g. [linear-algebra] or [matrices] for better visibility.

Linear transformation between finite dimensional vector spaces, $T:V\rightarrow W$, has a matrix representation $A_T$ with respect to given basis $B_V=\{v_1,\cdots,v_n\},\ B_W=\{w_1,\cdots,w_m\}$ Here we have another $A_T'$ with respect to basis $B_V'=\{v_1',\cdots,v_n'\},\ B_W'=\{w_1',\cdots,w_m'\}$ Also we have coordinate isomorphisms $P_V,\ P_W$ : $$ P_V v_i=v_i',\ P_W w_i=w_i' $$ So we have $P_WA_T'P_V^{-1}=A_T$