I'm working through a question that I don't quite know how to get.
It is: Find an inner product $\langle - , -\rangle$ on $\mathbb{R} ^3$, and a matrix A such that $\langle u, v \rangle = u^\top Av$ such that, with respect to $\langle - , -\rangle$, the basis $\biggl \{(1, 1, 0)^\top , (1, 0, 1)^\top , (0, 1, 1)^\top\biggr\} $ is orthonormal.
So, normally if I had an inner product that I was looking to find the orthonormal basis for I'd use the Gram-Schmidt process and divide by the norms. However, it doesn't seem as if you can use that process in reverse. I'm not looking for an answer in any way, more of how I'd get started on working this through.