Equivalent matrices represent the same linear transformation with resect to appropriate bases.

Theorem: Equivalent matrices represent the same linear transformation with respect to appropriate bases.

Explanation: If $A$ and $B$ are equivalent matrices (i.e., $B = PAQ$ for invertible $P$ and $Q$), then there exist bases $\alpha, \beta$ such that $A = [T]_{\alpha}^{\beta}$ and $B = [T]_{\alpha}^{eta}$ for the same linear transformation $T$.

Connection: Matrix equivalence captures the idea that the same abstract linear map looks different when expressed in different coordinate systems. The change-of-basis matrices $P$ and $Q$ account for the coordinate transformations in the domain and codomain.