How to find all the inner products that satisfy $\langle u_1,u_1 \rangle=1$ $\langle u_2,u_2 \rangle=1$? where $B:=\{u_1,u_2\}$ is a basis

Question

Let $V$ be a real vector space of $\dim 2$ and let $B:=\{u_1,u_2\}$ be a basis. How do you find all the inner products that satisfy $\langle u_1,u_1 \rangle=1$ $\langle u_2,u_2 \rangle=1$?

Answer 1

Pretty much anything will do, no? Namely, any inner product is determined by the values $\langle u_1,u_2\rangle,\langle u_1,u_1\rangle,\langle u_2,u_2\rangle$ and these values are independent. So, setting $\langle u_1,u_2\rangle=\alpha$ one sees that the inner product is given by $\langle au_1+bu_2,cu_1+du_2\rangle=ac+\alpha(ad+bu)+bd$