Let $V$ be n-dimensional vector space on $\mathbb R$, $(\cdot, \cdot)$ be an inner product on $V$, $\{a_1, \cdots , a_n\}$ be a basis of $V$.
And, let $\{ e_1, \cdots, e_n \}$ be an ONB of $V$, and let $P=(p_{ij})$ be a transformation matrix from $\{e_1, \cdots , e_n \}$ to $\{ a_1, \cdots , a_n\}$ i.e., $P$ satisfies $(a_1 \ a_2 \ \cdots \ a_n)=(e_1 \ e_2 \ \cdots \ e_n) P$.
(i) Let $B:=(b_{ij})$ where $b_{ij}=(a_i, a_j)$. Then, represent $B$ using $P.$
(ii) Prove that there exists $\{ u_1, \cdots , u_n \} \subset V$ s.t. $(a_i, u_j)=\delta_{ij}$.
So far, I solved (i). $B= ^t \! \!P P.$
For (ii), simply I let $u_j=\sum_{k=1}^n c_k^{(j)} a_k$ and I attempted to find $c_k^{(j)}$ s.t. $(a_i, u_j)=\delta_{ij}$.
i.e.,$(a_i, u_j)=(a_i, \sum_{k=1}^n c_k^{(j)} a_k)=\sum_{k=1}^n c_k^{(j)} (a_i, a_k)=\sum_{k=1}^n c_k^{(j)} b_{ik}$ and I want to determine $c_k^{(j)}$ so that $\sum_{k=1}^n c_k^{(j)} b_{ik}=\delta_{ij}$. (I couldn't find.) This method doesn't seem to work.
(And I also think $(a_1, \cdots , a_n)P^{-1}=(e_1, \cdots, e_n) $ is relative to $(a_i, u_j)=\delta_{ij}$.)
Thanks for any help.