Abstract
If θ ∈ (0,2π) is fixed, then the linear transformation
acts as a rotation of the plane ℝ2 by θ radians in the counterclockwise direction. For example, Rθ rotates the horizontal axis, namely, Spanℝ{e1}, to line
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.
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© 2001 Spinger-Verlag/New York, Inc
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Farenick, D.R. (2001). Invariant Subspaces. In: Algebras of Linear Transformations. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-0097-7_3
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DOI: https://doi.org/10.1007/978-1-4613-0097-7_3
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