Invariant Subspaces

  • Douglas R. Farenick
Part of the Universitext book series (UTX)


If θ ∈ (0,2π) is fixed, then the linear transformation
$$ R_\theta = \left( {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right) $$
acts as a rotation of the plane ℝ2 by θ radians in the counterclockwise direction. For example, R θ rotates the horizontal axis, namely, Span{e1}, to line
$$ L_\theta = Span_\mathbb{R} \left\{ {\left( {\begin{array}{*{20}c} {cos \theta } \\ {sin \theta } \\ \end{array} } \right)} \right\}. $$
One thing is clear about this simple linear transformation: because R θ is rotating lines that pass through the origin, the only value of θ ∈ (0,2π) for which R θ maps a line back into itself is θ = π. In this case, the rotation transformation is particularly simple, for its action on each vector v ∈ ℝ2 is just multiplication by the scalar -1: that is, R π v = −v for all v ∈ ℝ2.


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Copyright information

© Spinger-Verlag/New York, Inc 2001

Authors and Affiliations

  • Douglas R. Farenick
    • 1
  1. 1.Department of MathematicsUniversity of ReginaReginaCanada

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