  Despite its success, the composite fermion (CF) construction possesses some
mathematical features that have, until recently, not been fully understood. In
particular, it is known to produce wave functions that are not necessarily
orthogonal, or even linearly independent, after projection to the lowest Landau
level. While this is usually not a problem in practice in the quantum Hall
regime, we have previously shown that it presents a technical challenge for
rotating Bose gases with low angular momentum. These are systems where the CF
approach yields surprisingly good approximations to the exact eigenstates of
weak short-range interactions, and so solving the problem of linearly dependent
wave functions is of interest. It can also be useful for studying higher bands
of fermionic quantum Hall states. Here we present several ways of constructing
a basis for the space of so-called "simple" CF states for two-component
rotating Bose gases in the lowest Landau level, and prove that they all give
sets of linearly independent wave functions that span the space. Using this
basis, we study the structure of the lowest-lying state using so-called
restricted wave functions. We also examine the scaling of the overlap between
the exact and CF wave functions at the maximal possible angular momentum for
simple states.
