Suppose we have a linear operator $T:\Bbb F^n -> \Bbb F^n$. If we fix the basis as the standard basis for both the domain and the co-domain then it turns out that the matrix of an orthogonal transformation is an orthogonal matrix.
Q-1 Suppose I change the basis to a non standard basis, that is, something other than the standard basis, does the above still hold? That is to say, is the matrix of an orthogonal transformation an orthogonal matrix?
Q-2 Let $T: V \to W$ be a linear transformation where $V$ and $W$ can be any vector spaces. Fix bases for the domain and the co-domain, say $B$ and $C$. Does the concept of an orthogonal transformation still exist in this case because we would require the concept of a dot product and that the dot product must be preserved. In the case of $\Bbb F^n$ the vectors are column vectors with entries in the corresponding field and so the dot product can be defined. But for an arbitrary vector space the vectors are not column vectors although we can talk about the dot product of the coordinate vectors. So does such a thing as an orthogonal transformation exist for arbitrary spaces? If yes then does that mean the dot product $(X,Y)$ is preserved where $X$ and $Y$ are coordinate vectors?
Q-3 This is a continuation of Q-2. $T:V \to W$ is a linear transformation. Fix bases $B$ and $C$ for $V$ and $W$ respectively. Is the matrix with respect to the chosen bases orthogonal if the transformation is orthogonal?