I have been re-reading some linear analysis the last week, and I have been playing around a bit with the linear subspace of [tex]\mathbb{R}^\mathbb{R}[/tex] that you get with the basis {sin x, cos x}. It didn't take me long to realize that the family of transformations parametrized by theta [tex]T_\theta : \sin(x) \mapsto \sin(x + \theta)[/tex] (cos likewise) is a family of linear transformations on the space. However, when I looked at their matrices I was a bit surprised. W.r.t. the basis above, it is given by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]T_\theta = \left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right][/tex]

It can't be a coincidence that its matrix is a rotation matrix on [tex]\mathbb{R}^2[/tex]. Does anyone have a nice geometrical explanation of what is going on?

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# Geometry of linear operator

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