I am trying to understand why "naturality" is a good notion to consider in Category theory.
This brought me to reading a section of "General theory of natural equivalences" (https://pdfs.semanticscholar.org/3833/f692e93a795b51bd3d7acfd9477f9ee6d536.pdf) and was struggling with the comment given at the really bottom of page 235.
Why is it that it is not possible to have a non-singular bilinear form which is invariant under any non-singular linear transformation?
If you have a vector space $V$ of dimension $\ge2$ and a nonsingular bilinear form $B$, there will be a pair of linearly independent vectors $u_1$ and $u_2$ with $B(u_1,u_2)=0$ and there will be a pair of linearly independent vectors $v_1$ and $v_2$ with $B(v_1,v_2)=1$.
There will be a nonsingular map on $V$ taking $u_i$ to $v_i$. That map cannot possibly preserve $B$.