(a) Find the inverse of the rotation matrix Rθ=( cosθ -sinθ sinθ cosθ ), where θ∈ℝ.

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(a) Find the inverse of the rotation matrix Rθ=( cosθ ...

(a) Find the inverse of the rotation matrix $R_\theta=\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$, where $\theta \in \mathbb{R}$. (b) Use your result to solve the system $x=a \cos \theta-b \sin \theta, y=a \sin \theta+b \cos \theta$, for $a$ and $b$ in terms of $x$ and $y$. (c) Prove that, for all $a \in \mathbb{R}$ and $0<\theta<\pi$, the matrix $R_\theta-a$ I has an inverse.

(a) Find the inverse of the rotation matrix $R_\theta=\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$, where $\theta \in \mathbb{R}$.
(b) Use your result to solve the system $x=a \cos \theta-b \sin \theta, y=a \sin \theta+b \cos \theta$, for $a$ and $b$ in terms of $x$ and $y$. (c) Prove that, for all $a \in \mathbb{R}$ and $0<\theta<\pi$, the matrix $R_\theta-a$ I has an inverse.

Key Concepts

Rotation Matrix

A rotation matrix is a special type of square matrix used to perform a rotation in Euclidean space. In two dimensions, it is defined by an angle ? and rotates any vector it multiplies by that angle around the origin. Its structure, based on sine and cosine functions, ensures that it preserves distances and angles, making it an example of an isometry.

Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns (and rows) are orthonormal. One of the key properties of orthogonal matrices is that their inverse is equal to their transpose. This property simplifies the process of computing the inverse of rotation matrices and underpins many of their useful geometric features.

Matrix Inversion

Matrix inversion is the process of finding a matrix that, when multiplied with the original matrix, results in the identity matrix. For rotation matrices, due to their orthogonal nature, the inverse can be directly obtained as the transpose, which is a simpler and more efficient computation compared to inverting a general matrix.

Systems of Linear Equations

Systems of linear equations often arise in the context of transformations applied to coordinate systems. When a transformation, such as a rotation, is represented by a matrix, solving the system involves applying the inverse transformation. This operation is typically carried out by multiplying both sides of the equation by the inverse of the transformation matrix, yielding the solution in terms of the original variables.

Invertibility Conditions

The invertibility of a matrix, such as the one given by R? - aI, hinges on the matrix having a nonzero determinant. In the context of parameterized matrices, conditions on the parameters (like a and the rotation angle ?) ensure that the matrix does not become singular, thus guaranteeing the existence of an inverse.

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