If we let $\hat{\rho}$ be the density matrix of a Bell state. I have seen that calculating the trace,
$$ Tr(\hat{A} \otimes \hat{I} \hat{\rho}) = Tr_{\mathcal{H}_A}(\hat{A}\hat{\rho}_{\mathcal{H}_A}) $$
Where $Tr_{\mathcal{H}_A}$ is the partial trace in the Hilbert space $\mathcal{H}_A$ and $\hat{\rho}_{\mathcal{H}_A}$ is the reduced density matrix $\hat{\rho}_{\mathcal{H}_A} = Tr_{\mathcal{H}_B}(\hat{\rho})$.
Would someone please be able to provide some clarity on why this is true, and if there is a more general formula for when we act on $\hat{\rho}$ with an arbitrary linear operator $\hat{A}\otimes\hat{B}$?
