Math Problem Statement

What is the matrix of the rotation R_t : R^2 → R^2 by t radians anticlockwise?

Solution

The question asks for the matrix of a rotation RtR_t that rotates vectors in R2\mathbb{R}^2 by tt radians in the anticlockwise direction.

The general form of a 2D rotation matrix for a rotation by tt radians anticlockwise is:

\cos(t) & -\sin(t) \\ \sin(t) & \cos(t) \end{pmatrix}$$ So, the matrix $$R_t$$ has entries involving trigonometric functions based on the angle $$t$$. Would you like a more detailed explanation of how this matrix is derived? Here are some related questions for further practice: 1. What would the matrix look like for a clockwise rotation by $$t$$ radians? 2. How would the matrix change if we were in $$\mathbb{R}^3$$ instead of $$\mathbb{R}^2$$? 3. What is the inverse of the rotation matrix $$R_t$$, and what does it represent? 4. How would you apply the matrix $$R_t$$ to rotate a specific vector in $$\mathbb{R}^2$$? 5. What happens to the determinant of the matrix $$R_t$$ for any value of $$t$$? **Tip**: The determinant of any 2D rotation matrix is always 1, which reflects that rotations preserve the area and orientation of figures.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
2D Rotation
Trigonometry

Formulas

Rotation Matrix R_t = [[cos(t), -sin(t)], [sin(t), cos(t)]]

Theorems

Rotation Matrix in 2D
Properties of Orthogonal Matrices

Suitable Grade Level

Undergraduate Level (Linear Algebra)