The reason behind the name "Orthogonal transformation". An orthogonal transformation is a linear transformation such that $(Tx,Ty)=(x,y)$. 
Orthogonality is suggestive of perpendicularity. What might have been the reason for naming a distance preserving linear transformation on a vector space "orthogonal"?
Thanks!

Over a finite dimensional vector space, the following are equivalent:


*

*$T$ preserves inner products (i.e. angles): $\langle Tx,Ty\rangle=\langle x,y\rangle$ for each $x,y$.

*$T$ preserves orthonormal bases: For every orthonormal basis $B$, $T(B)$ is an orthonormal basis.

*$T$ preserves an orthonormal basis: There exists an orthonormal basis $B$ such that $T(B)$ is an orthonormal basis.

*$T$ preserves distances $\lVert Tx\rVert =\lVert x\rVert$ for each $x$.

*$TT^\ast=T^\ast T=1$


They all define what we call an orthogonal transformation, one that preserves all geometrical relations in our space.
