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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 $\mathbb{R}^{n\times 1}$?

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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

$$(x_1,\dots,x_n)\cdot (y_1,\dots,y_n):=x_1y_1+\dots+x_ny_n.$$

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Orthogonality is always with respect to a given inner product. Changing the inner product will change whether a transformation is orthogonal or not. So if you are interested in orthogonality with respect to the standard inner product, then you only need to check with the standard inner product.

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