from Irving Kaplansky's "Linear Algebra and Geometry a second course." Page 19.
Theorem: Let $V$ be a nonsingular two dimensional inner product space over a field of characteristic not equal to $2$, and assume that $V$ contains a vector $x$ such that $(x,x)=0$. Then we can diagonalize $V$ so that the matrix of it's inner product has diagonal entries $(1,-1)$.
The author goes on to write that that an inner product space $V$ is called a hyperbolic plane if it satisfies the hypothesis of the theorem above.
This definition of a hyperbolic plane in terms of the existence of a null vector is new to me. My question is meant to be open ended and not too specific, but I was wondering if anyone could expand on this definition of the hyperbolic plane, and really just give me any insights what-so-ever. Thank you!