# Prove that a function is an orthogonal transformation

I want to prove that a function $$f\colon \mathbb{R}^{n\times 1} \to \mathbb{R}^{n\times 1}$$ is an orthogonal transformation on $$\mathbb{R}^{n\times 1}$$.

Following this article from wikipedia, a function $$T\colon V\to V$$ on an inner product space $$V$$ is an orthogonal transformation if $$\langle u,v\rangle=\langle T(u),T(v)\rangle, \quad \forall u,v\in V.$$

My question is, if it is sufficient for my function $$f$$ to be an orthogonal transformation if I prove this property only for the standard euclidean inner product, or do I have to prove this property for any arbitrary inner product on $$\mathbb{R}^{n\times 1}$$?

When talking about orthogonal transformation of a vector space $$E$$, it is understood that $$E$$ has a canonical inner product (this is the same thing when you say $$E$$ is a vector space: it must have canonical vector addition and scalar multiplication, or otherwise you have to precise $$(E,+_E,\cdot_E)$$). When talking about $$\mathbb{R}^n$$ as an inner product space, it is understood that it is for the canonical inner product
$$(x_1,\dots,x_n)\cdot (y_1,\dots,y_n):=x_1y_1+\dots+x_ny_n.$$