Linear transformations - rotations
Linear transformations - rotations
Understanding Linear Transformations: Rotations
Basic Concept of Rotations
- A rotation in Linear Algebra is a type of linear transformation that changes the positioning of points in a plane while keeping its orientation.
- The key elements of a rotation are the centre of rotation and the angle of rotation, both of which are typically provided in the problem.
- Rotation transformations are defined by 2-dimensional square matrices.
Matrix Representation of Rotations
- Rotations in the plane are represented by 2x2 matrices.
- For a counterclockwise rotation by an angle θ, the transformation matrix is
[cos(θ), -sin(θ); sin(θ), cos(θ)]. - For a clockwise rotation, the signs of the sine terms are reversed:
[cos(θ), sin(θ); -sin(θ), cos(θ)].
Applying a Rotation Matrix
- To apply a rotation, you multiply the rotation matrix by the column vector representing the original point.
- Remember that in matrix multiplication, the order matters. The rotation matrix should be on the left and the original point column vector on the right.
- This results in a new column vector that represents the position of the point after the rotation.
Features of Rotation Matrices
- An important feature of rotation matrices is that their determinant is always equal to 1.
- Rotation matrices are also orthogonal. In other words, the dot product of each row (or column) with itself is 1, and the dot product of different rows (or columns) is 0.
Rotations and Complex Numbers
- Every rotation in the plane can be understood as multiplying by a complex number.
- Specifically, a counterclockwise rotation by an angle θ is the same as multiplying by the complex number
cos(θ) + isin(θ).
Reacting to Problem Scenarios
- Always be attentive to the direction of rotation (clockwise or counterclockwise) and ensure this is aligned with the matrix used.
- The centre of the rotation and scale of the rotation (angle) should be clearly understood before beginning any calculations.
Contextual Applications
- Rotation transformations are widely utilized in the fields of computer graphics and robotic movements.
- They are central in modelling rotations in physical systems in physics and engineering.
Review and Preparation
- Always cross-check the output of the rotation against the expected direction and scale of the rotation.
- Constantly practise with an array of problems to fully understand and remember the rotation matrices and their application.