This question may not be of theoretical importance in Linear algebra, but I came to this question, while looking definition of orthogonal transformation in intuitive way.
Let $V$ be an inner product space over $\mathbb{R}$. A linear transformation $T\colon V\rightarrow V$ is called orthogonal if $<T(v),T(w)>=<v,w>$ for all $v,w\in V$.
A basis $B=\{v_1,v_2,\cdots, v_n\}$ of $V$ is called orthogonal if $<v_i,v_j>=0$ for $i\neq j$.
A basis $B=\{v_1,v_2,\cdots, v_n\}$ of $V$ is called orthonormal if $<v_i,v_j>=$ for $i\neq j$ and $<v_i,v_i>=1$.
Now it is a
Theorem: $T\colon V\rightarrow V$ is orthogonal if and only if $T$ takes an orthonormal basis to orthonormal basis.
I initially confused with this theorem; I thought that $T$ is orthogonal transformation if and only if $T$ takes orthogonal basis to orthogonal basis. (This is wrong statement).
But, the intuitive way to define an orthogonal transformation would be the bold faced statement.
Can anybody point out the historical references for the definition of orthogonal transformation and the Theorem 1 mentioned above?
The issue is the historical use of "orthogonal" for isometrics of the space. It's somewhat understandable to want them called orthonormal transformations, but it seems that transformations that just preserve orthogonality of bases are just not common enough to have a name. They include conformal linear transformations, but I'm not sure offhand if they characterize them.
For a highly related discussion, see this: Why is an orthogonal matrix called orthogonal?