In linear algebra, rotation refers to a specific type of linear transformation that turns a vector or a point around a fixed point (usually the origin) without changing its size or shape. This transformation is fundamental for describing movement and orientation in space.
Understanding Rotation Through Transformation Matrices
The primary tool for performing rotations in linear algebra is the rotation matrix. As stated in the reference, "In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space."
Think of a transformation matrix as a mathematical instruction set that tells you how to move points or vectors. A rotation matrix specifically contains the information needed to rotate them by a certain angle about a fixed point.
How Rotation Matrices Work
When you multiply a vector (representing a point or direction) by a rotation matrix, the result is a new vector that represents the original point or vector after it has been rotated.
This process can be applied to single points, entire shapes defined by sets of points (like the vertices of a polygon), or even basis vectors that define a coordinate system.
Example: 2D Rotation About the Origin
A common and illustrative example is rotation in a two-dimensional (2D) plane, such as the xy-plane.
According to the reference, using a standard convention: "the matrix rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system."
The specific 2D rotation matrix for a counterclockwise rotation by an angle θ about the origin is:
| x' | y' | |
|---|---|---|
| x | cos(θ) | -sin(θ) |
| y | sin(θ) | cos(θ) |
(Note: The matrix structure is represented here, typically written as):
[ cos(θ) -sin(θ) ]
[ sin(θ) cos(θ) ]If you have a point with coordinates (x, y), you can find its new coordinates (x', y') after rotating by θ using matrix multiplication:
[ x' ] [ cos(θ) -sin(θ) ] [ x ]
[ y' ] = [ sin(θ) cos(θ) ] [ y ]
This multiplication results in:
x' = x * cos(θ) - y * sin(θ)y' = x * sin(θ) + y * cos(θ)
This formula precisely describes how the x and y coordinates change when rotated counterclockwise by angle θ.
Key Aspects of Rotation in Linear Algebra
- Linear Transformation: Rotation is a linear transformation because it satisfies properties like preserving vector addition and scalar multiplication.
- Preserves Distance: Rotations are rigid transformations; they don't stretch or shrink objects.
- Preserves Angle: The angles between vectors or lines remain unchanged after rotation.
- Determined by Angle and Axis/Point: In 2D, it's determined by an angle and a point of rotation (usually the origin). In 3D, it's determined by an angle and an axis of rotation passing through the origin.
- Represented by Matrices: Rotation matrices provide a concise and powerful way to apply rotation transformations mathematically.
In summary, rotation in linear algebra is the geometric transformation of turning points or vectors, implemented mathematically through multiplication by a specific type of transformation matrix known as a rotation matrix.
