Invariant Subspaces

Abstract

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_ℝ{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_πv = −v for all v ∈ ℝ².