Find an inner product on $\mathbb{R}^2$

Question

Find an inner product on $\mathbb{R}^2$ such that $\langle e_1,e_2\rangle =2$. I don't understand how can I find it. Help.

Answer 1

Consider the linear map $T: \mathbb R^2 \to \mathbb R^2$, where $T(e_2)=2e_1+e_2$ and $T(e_1)=e_1$. Then $T$ is an automorphism, and define

$(v,w)=T(v) \cdot T(w)$ where $\cdot$ is the usual dot product.