Yes, a rotation matrix represents a linear transformation.
Understanding Linear Transformations
A transformation (or mapping) $T$ is considered linear if it satisfies two fundamental properties for any vectors $\mathbf{u}$, $\mathbf{v}$ in its domain and any scalar $c$:
- Additivity (Superposition Principle): $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$
- Homogeneity (Scalar Multiplicativity): $T(c\mathbf{v}) = cT(\mathbf{v})$
Geometrically, a linear transformation always maps the origin to the origin ($T(\mathbf{0}) = \mathbf{0}$), and it maps straight lines to straight lines (or points). It preserves the operations of vector addition and scalar multiplication.
Why Rotation Matrices Are Linear
A rotation matrix, denoted as $R$, transforms a vector $\mathbf{v}$ into a new vector $R\mathbf{v}$ by rotating it around a fixed point (typically the origin). This operation precisely meets the criteria for a linear transformation.
Here's why:
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Matrix Multiplication Properties: The core of a rotation is matrix-vector multiplication. Matrix multiplication inherently satisfies the properties of additivity and homogeneity.
- Additivity: When you apply a rotation to the sum of two vectors, $R(\mathbf{u} + \mathbf{v})$, matrix algebra dictates that this is equivalent to rotating each vector separately and then adding the results: $R\mathbf{u} + R\mathbf{v}$.
- Homogeneity: Similarly, rotating a scaled vector, $R(c\mathbf{v})$, is the same as scaling the vector first and then rotating it, or rotating the vector first and then scaling the result: $c(R\mathbf{v})$.
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Fixed Origin: A pure rotation always keeps the origin stationary. If you rotate the zero vector, it remains the zero vector. This is a characteristic feature of linear transformations.
Addressing the "Trigonometric Entities" Point
It's common to wonder why rotation matrices are considered linear given that their elements are derived from trigonometric functions like sine and cosine. This is a valid point of confusion, as noted in the internal reference.
The key is that the trigonometric functions (sin, cos) are used to define the constant entries within the rotation matrix itself for a given rotation angle. They are not functions applied to the input vector's components during the transformation.
For example, a 2D rotation matrix by an angle $\theta$ is:
$R = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix}$
When you apply this to a vector $\mathbf{v} = \begin{pmatrix} x \ y \end{pmatrix}$, the transformation is:
$T(\mathbf{v}) = R\mathbf{v} = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} x\cos\theta - y\sin\theta \ x\sin\theta + y\cos\theta \end{pmatrix}$
Here, $\cos\theta$ and $\sin\theta$ are just fixed coefficients. The variables $x$ and $y$ (the components of the input vector) appear in a linear combination, not inside the trigonometric functions themselves. This structure ensures that the transformation satisfies additivity and homogeneity.
Mathematical Proof of Linearity
Let $R$ be a rotation matrix.
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Additivity:
$R(\mathbf{u} + \mathbf{v}) = R\mathbf{u} + R\mathbf{v}$
This property holds true due to the distributive property of matrix multiplication over vector addition. -
Homogeneity:
$R(c\mathbf{v}) = c(R\mathbf{v})$
This property holds true because scalar multiplication can be factored out of matrix-vector products.
These two properties rigorously confirm that any rotation represented by a matrix is indeed a linear transformation.
Distinguishing Rotation from General Affine Transformations
The internal reference also mentions affine transformations. It's crucial to understand the distinction:
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Rotation Matrices: Are a subset of affine transformations. Specifically, they are linear transformations that also preserve lengths and angles (making them a type of rigid transformation). They always keep the origin fixed.
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General Affine Transformations: These are transformations that preserve collinearity (points on a line remain on a line) and ratios of distances along a line. They can include operations like scaling, shear, reflection, rotation, and translation.
- A general affine transformation (e.g., one that includes a translation) is not linear in the strict sense because it typically does not map the origin to the origin ($T(\mathbf{0}) \neq \mathbf{0}$).
- However, affine transformations are often represented as linear transformations in a higher-dimensional space using homogeneous coordinates. For example, a 2D translation can be represented by a 3x3 matrix acting on 3D homogeneous coordinates. In this higher-dimensional space, the operation is linear. But in the original 2D space, the translation itself is not.
Because a pure rotation matrix (without any translation component) always maps the origin to the origin and satisfies both linearity conditions, it is unequivocally a linear transformation.
| Property | Linear Transformation (e.g., Rotation) | Non-Linear Transformation (e.g., Translation, non-linear function) |
|---|---|---|
| Additivity | Preserved: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ | Not necessarily preserved |
| Homogeneity | Preserved: $T(c\mathbf{v}) = cT(\mathbf{v})$ | Not necessarily preserved |
| Origin Mapping | Origin maps to origin: $T(\mathbf{0}) = \mathbf{0}$ | Origin may map to a different point ($T(\mathbf{0}) \neq \mathbf{0}$ for translation) |
| Line Mapping | Lines map to lines (or points) | Lines may map to curves (e.g., $y=x^2$) |
Practical Applications and Importance
The linearity of rotation matrices is fundamental in numerous fields:
- Computer Graphics: Used extensively for rotating objects, cameras, and coordinate systems in 2D and 3D environments.
- Robotics: Essential for controlling robot arm movements and specifying orientations in space.
- Physics and Engineering: Applied in mechanics, kinematics, and dynamics to describe the orientation and motion of rigid bodies.
- Machine Learning: Used in data augmentation (rotating images) and certain feature transformations.
Understanding that rotations are linear transformations simplifies their mathematical treatment and allows for powerful computational techniques based on linear algebra.
