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Exercise 9

Avg difficulty 3.0

For each real number θ\theta let tθ ⁣:R2→R2t_{\theta}\colon\mathbb{R}^{2}\to\mathbb{R}^{2} be represented with respect to the standard bases by this matrix.

(cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}

Show that tθ1+θ2=tθ1⋅tθ2t_{\theta_{1}+\theta_{2}}=t_{\theta_{1}}\cdot t_{\theta_{2}}. Show also that tθ−1=t−θ{t_{\theta}}^{-1}=t_{-\theta}.

Hefferon, Linear Algebra, Maps Between Spaces, Inverses, Exercise 9

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.