You can do it with `FindGeometricTransform[]`:

    data = {{13.2, 200, 58, 21.2}, {10, 263, 48, 44.5}, {8.1, 294, 80, 31}, 
            {8.8, 190, 50, 19.5}, {9, 276, 91, 40.6}, {7.9, 204, 78, 38.7}, 
            {3.3, 110, 77, 11.1}, {5.9, 238, 72, 15.8}, {15.4, 335, 80, 31.9}, {17.4, 211, 60, 25.8}};
    {error, f} = FindGeometricTransform[PrincipalComponents[data], data];
    Norm[PrincipalComponents[data][[1]] - f[data[[1]]]]
    (*
      0.8
    *)