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Two representations ρ: G → GL(V) and σ: G → GL(W) are equivalent (isomorphic) if there exists an invertible linear map T: V → W that intertwines the two actions: T∘ρ(g) = σ(g)∘T for all g ∈ G. In matrix terms, this means ρ and σ differ by a change of basis. Equivalence is the right notion of "sameness" for representations — it preserves all structural properties while ignoring coordinate artifacts.
In any mathematical theory, once you define objects, the next essential step is to define when two objects are "the same." For representations, the right notion is equivalence (also called isomorphism). Two representations ρ: G → GL(V) and σ: G → GL(W) are equivalent if there exists an invertible linear map T: V → W such that T∘ρ(g) = σ(g)∘T for all g ∈ G. The map T is called an intertwining isomorphism or a G-isomorphism.
The intertwining condition T∘ρ(g) = σ(g)∘T says that it does not matter whether you first apply the G-action and then translate via T, or first translate via T and then apply the G-action. In matrix terms, if ρ and σ are both n-dimensional, the condition becomes Tρ(g) = σ(g)T, or equivalently σ(g) = Tρ(g)T⁻¹ for all g — which is exactly conjugation by T. So equivalent matrix representations are related by a single change of basis applied uniformly to all group elements.
Why does this definition matter? Because many superficially different-looking representations are secretly the same. The rotation group SO(2) acting on ℝ² in the standard basis gives the familiar rotation matrices. In an eigenvector basis (over ℂ), the same action becomes diagonal. These look different as matrices but carry identical structural information — they are equivalent representations. The goal of representation theory is to classify representations up to equivalence, stripping away coordinate noise to reveal the underlying structure.
The intertwining operators that are not necessarily invertible also play a crucial role. The set Hom_G(V, W) of all G-equivariant linear maps from V to W forms a vector space, and its dimension measures how "similar" two representations are. Schur's lemma, which you will encounter soon, shows that when V and W carry irreducible representations, this space is either zero-dimensional (the representations are inequivalent) or one-dimensional (they are equivalent). This is the beginning of a systematic classification program.