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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def expand_matrix_in_orthogonal_basis( m: np.ndarray, basis: Dict[str, np.ndarray], ) -> value.LinearDict[str]: """Computes coefficients of expansion of m in basis. We require that basis be orthogonal w.r.t. the Hilbert-Schmidt inner product. We do not require that basis be orthonormal. Note that Pauli basis (I, X, Y, Z) is orthogonal, but not orthonormal. """

    return value.LinearDict({
        name: (hilbert_schmidt_inner_product(b, m) /
               hilbert_schmidt_inner_product(b, b))
        for name, b in basis.items()
    })