= Dual isomorphisms induced by a nondegenerate pairing
{title2=$U\cong V^*,\quad V\cong U^*$}
A <bilinear form> $b:U\times V\to K$ with zero left and right radicals induces injective <linear maps> $U\to V^*$ and $V\to U^*$. Finite dimensions force $\dim U=\dim V$, so both maps are <isomorphisms>. For any other bilinear form $c$, composing its maps to the <dual spaces> with these inverse <isomorphisms> gives unique <endomorphisms> $S,T$ with $c(x,y)=b(Sx,y)=b(x,Ty)$. In bases with invertible pairing <matrix> $B$ and coefficient <matrix> $C$, $S=B^{-T}C^T$ and $T=B^{-1}C$.
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