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My text (p. 19) introduces the concept of a "rotated" basis without explanation. What properties or characteristics of a basis make it "rotated" with respect to another? What operation on one basis produces a basis that is rotated with respect to it?

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In this context, it means they differ by a unitary transformation of $\mathbb{C}^n$.

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Is there a more elementary way to state that, or break it down? It seems to me (linear algebra is a distant and dimming memory; I'm not sure what "unitary transformation" means, for example) that if you start with a set of "orthonormal" vectors, $S$ in $\mathbb{C}^n$, then any other such set of vectors will be "rotated" with respect to $S$. Is that right? –  raxacoricofallapatorius Feb 23 '13 at 19:45
    
Correct, any 2 orthonormal bases differ by a unitary transformation. This follows directly from the definition of unitary: $UU^* = I$, where $U^*$ is the conjugate transpose of $U$. –  Ted Feb 23 '13 at 19:48
    
So in contexts like my text, "rotated" pretty much means "any", since we're generally assuming "basis" $\equiv$ "orthonormal basis"? –  raxacoricofallapatorius Feb 23 '13 at 19:51
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Yes, that's correct. –  Ted Feb 23 '13 at 19:56
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Yes. Unlike the case of the orthogonal group over $\mathbb{R}$, which splits into "rotations" of determinant 1 and "reflections" of determinant -1, the unitary group over $\mathbb{C}$ is connected, so there is no hard split. The determinant of a unitary matrix can be any number $e^{i \theta}$ on the unit circle. However, you could limit yourself to the special unitary transformations of determinant 1, in certain contexts. This would exclude the "stretching" unitary transformations like multiplication by $e^{i \theta}$. –  Ted Feb 24 '13 at 1:25

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