= Exact factor pricing without idiosyncratic risk
{title2=$a-R\mathbf1\in\operatorname{rowspan}B$}
For returns $r_i=a_i+\sum_jb_{ji}f_j$, fund a risky position $v$ by a <risk-free asset>. Its zero-cost payoff is $(a-R\mathbf1)^Tv+\sum_j(Bv)_jf_j$. If $Bv=0$, a nonzero constant term would create an <arbitrage> after choosing the sign of $v$. Hence $a-R\mathbf1$ annihilates $\ker B$ and lies in the row span of $B$. Taking <expectations> gives exact factor-premium pricing. Full row <rank> makes the coefficients unique; if exposures have smaller <rank>, only the effective spanned factors are identified.
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