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Can someone point me to a good explanation of the intrinsic definition of an orthogonal/unitary transformation? By this I mean one which does not make reference to matrices or matrix operations like the transpose.

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Let $(V, \cdot )$ be an inner product space and $T$ an endomorphism of $V$. The adjoint $T^{*}$ of $T$ is the endomorphism of $V$ defined as follows: $T(v) \cdot w=v \cdot T^{*}(w)$ for all $v,w \in V$. Then we say the linear transformation $T: V \rightarrow V$ is orthogonal if $TT^{*}=id$, where $id$ is the identity map on $V$.

It follows easily from this definition that an orthogonal transformation preserves the inner product (in fact many define an orthogonal transformation as an endomorphism which preserves the inner product!).