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$$ \widetilde{I}' = \Lambda \widetilde{I} \Lambda ^{\dagger}$$

Now, if one were to apply some vector stimulus in the primed coordinate system from the right, would it be correct to think of this vector as firstly being transformed into the unprimed coordinate system (since the adjoint is equivalent to the inverse within this context), then being directionally altered by the inertia tensor in the unprimed coordinate system, then finally being transformed back into the primed coordinate system by the final matrix? I feel like I'm misunderstanding something fundamental however.

Thanks in advance,

-Vulcan