Types of Transformation Matrices

Previous: Types of Linear Transformations Next:

Earlier, we introduced rotation matrices; matrices that rotate vectors in the coordinate plane. Now we discuss them in detail, as well as other types of transformation matrices.

Rotation Matrices

Let $R(\theta ) $ be a rotation matrix that rotates vectors by $\theta $ radians from the $x$-axis, and that a vector $\vec{v} = \begin{bmatrix} x \\ y \end{bmatrix}$ is transformed to the point $\vec{v'} = \begin{bmatrix} x' \\ y' \end{bmatrix}$, with $\phi$ as the angle between the $x$-axis and the segment from the origin to the head of the vector, as shown below: Using trigonometric ratios, it can be seen that $$x = r\cos \phi \text{ and } y = r\sin \phi $$ $$x' = r\cos \theta \text{ and } y' = r\cos \theta.$$ By the addition formulae in trigonometry, it follows that $$x' = r\cos(\theta + \phi) = r\cos\theta \cos\phi - r\sin\theta \sin\phi$$ and $$y' = r \sin(\theta + \phi) = r\sin\theta \cos\phi + r\cos\theta \sin\phi $$.

$R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix}$