Let V be a real two-dimensional vector space with basis $\{e_1, e_2\}$. Find all the inner products $\langle$–,–$\rangle$ on V which satisfy $\langle e_1, e_1 \rangle = \langle e_2, e_2 \rangle = 1$
I'm able to mess around with a lot of the properties of this, but I really can't figure out how at all to get the required result.
The general form of an inner product on $\mathbb{R}^n$ is given by $$ \langle x,y\rangle = y^tAx, $$ where $A$ denotes a real positive-definite, symmetric matrix of size $n$. For $n=2$ we can apply Hurwitz criterion, i.e., we obtain the matrices $$ A=\begin{pmatrix} a & b \\ b & c \end{pmatrix} $$ with $a>0$ and $ac-b^2>0$.
Edit: The question has the additional requirement that the inner product is normalised. This means $1=\langle e_1,e_1\rangle =e_1^tAe_1=a$, and $1=\langle e_2,e_2\rangle =e_2^tAe_2=c$.