Given a basis $U=(u_1,cdots,u_n)$ not necessarily orthonormal, in $scrH$, how would you compute the biorthogonal basis $sex{v_1,cdots,v_n}$? Find a formula that expresses $sef{v_j,x}$ for each $xinscrH$ and $j=1,cdots,k$ in terms of Gram matrices.
Soluton. Let $V=(v_1,cdots,v_k)$, then $$ex V^*U=I_nlra U^*V=I_n. eex$$ We may just set $v_i$ to be the solution of the linear system $U^*x=e_i$, where $e_i=(underbrace{0,cdots,1}_{i},cdots, 0)^T$. Suppose now $$ex x=sum_{j=1}^n x_jv_jin scrH, eex$$ then $$ex sef{v_i,x}=sum_{j=1}^n sef{v_i,v_j}x_j,quad i=1,cdots,n. eex$$ And hence $$ex sex{a{cc} sef{v_1,x}\ vdots\ sef{v_n,x} ea}=sex{sef{v_i,v_j}}sex{a{cc} x_1\vdots\ x_n ea}. eex$$