A transformation is orthogonal if it preserves a symmetric inner product. As a consequence, the lengths of vectors and the angles between vectors are preserved.
Symmetric means <u,v>=<v,u>.
Appendix A – Positive Definite
<v,v> is positive except for the case where v=0; <0,0>=0
Metric spaces is subset of topological spaces. Normed Vector spaces is subsets of metric spaces. Inner product spaces is subset of normed vector spaces.
We should expect additional rules to make things more restrictive as we go from set to subset.
All Inner Product Spaces are Vector Spaces.
Vectoreverything 