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First published on Thursday, Jan 9, 2025 and last modified on Thursday, Jan 9, 2025
Mathedu
Linear Mapping, Matrix, Rotation
In that test, you will prove some results about the rotations with the help of their matrices in the canonical base.
Consider the two following rotations and there matrices.
The rotation \( \rho_{\theta}\) of angle \( \theta\) , having the matrix \( R_{\theta}=\begin{bmatrix} \cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{bmatrix}\) in the canonical base.
And the rotation \( \rho_{\mu}\) of angle \( \mu\) , having the matrix \( R_{\mu}=\begin{bmatrix} \cos(\mu)&-\sin(\mu)\\ \sin(\mu)&\cos(\mu)\end{bmatrix}\) in the canonical base.
Use the trigonometric formulae to prove that \( R_{\theta}R_{\mu}=R_{(\theta+\mu)}\) .
Deduce that \( \rho_{\theta}\circ\rho_{\mu}=\rho_{(\theta+\mu)}\) .
Consider the following rotation and its matrix.
The rotation \( \rho_{\theta}\) of angle \( \theta\) , having the matrix \( R_{\theta}=\begin{bmatrix} \cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{bmatrix}\) in the canonical base.
Use the trigonometric fundamental formula to prove that \( \det(R_{\theta})=1\) .
Deduce that \( R_{\theta}\) and \( \rho_{\theta}\) , are invertible.
Use the parity of the cosine and sine to prove that \( R_{\theta}^{-1}= R_{-\theta}\) .
Deduce that \( \rho_{\theta}^{-1}=\rho_{-\theta}\) .