Let $X$ be nay basis of $scrH$ and let $Y$ be the basis biorthogonal to it. Using matrix multiplication, $X$ gives a linear transformation from $bC^n$ to $scrH$. The inverse of this is given by $Y^*$. In the special case when $X$ is orthonormal (so that $Y=X$), this transformation is inner-product preserving if the standard inner product is used on $bC^n$. eex$$
解答: $$eex ea sex{a{c} a_1\ vdots\ a_n ea}inbC^n& a Xsex{a{c} a_1\ vdots\ a_n ea}in scrH,\ Xsex{a{c} a_1\ vdots\ a_n ea}=sex{a{c} b_1\ vdots\ b_k ea}& a sex{a{c} a_1\ vdots\ a_n ea}=Y^*sex{a{c} b_1\ vdots\ b_k ea},\ sef{Xsex{a{c} a_1\ vdots\ a_n ea},Ysex{a{c} b_1\ vdots\ b_n ea}}&=sex{ar a_1,cdots,ar a_n}X^*Ysex{a{c} b_1\ vdots\ b_n ea}=sum_{i=1}^n ar a_ib_i. eea eeex$$