# Invariant Subspaces

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

## 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, Rθ 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 Rθ is rotating lines that pass through the origin, the only value of θ ∈ (0,2π) for which Rθ 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, Rπv = −v for all v ∈ ℝ2.

## Keywords

Irreducible Representation Linear Transformation Invariant Subspace Left Ideal Division Algebra
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.