Given two matrices $A$ and $B$, they are similar if: $$B=P^{-1}AP $$
Furthermore, if they are similar they are relative to the same linear transformation (equivalent). However the proof I've checked for this, coming from ProofWiki, essentially simply defines equivalence (in two ways, actually) as $B=Q^{-1}AP$ and does $Q=P$. In this way, it's obvious that similarity implies equivalence, however I don't get why that definition of equivalence means "what we want it to mean": that the two matrices represent the same linear transformation. Can you help me understand?