Final answer:
To find the matrix of the linear transformation that rotates any vector through an angle of theta in the clockwise direction, we can use the rotation matrix in two dimensions. By arranging the components of the image vectors as columns of the matrix, we can obtain the matrix of the linear transformation.
Step-by-step explanation:
To find the matrix of the linear transformation that rotates any vector through an angle of θ in the clockwise direction, we can use the following steps:
- Using the rotation matrix in two dimensions, which is given by:
- R = [cos(θ) -sin(θ)]
- [sin(θ) cos(θ)]
Identify the dimensions of the input and output vectors. Since the transformation is happening in 2D, the input vector will have two components and the output vector will also have two components.To get the matrix of the linear transformation, we arrange the components of the image vectors as columns of the matrix. So, the matrix of the linear transformation is:
- T = [cos(θ) -sin(θ)]
- [sin(θ) cos(θ)]