1. The problem asks why a transformed vector is a scalar multiple of the vector representing the twinned line. 2. When a vector is transformed by a linear transformation and the result is a scalar multiple of the original vector, it means the vector is an eigenvector of the transformation. 3. The scalar multiple is called an eigenvalue, and the vector itself is called an eigenvector. 4. This relationship can be expressed as $$T(\mathbf{v}) = \lambda \mathbf{v}$$ where $T$ is the transformation, $\mathbf{v}$ is the vector, and $\lambda$ is the scalar (eigenvalue). 5. In the context of a twinned line, the vector representing the line remains in the same direction after transformation, only scaled by $\lambda$. 6. This means the transformation stretches or compresses the vector but does not change its direction, which is why the transformed vector is a scalar multiple of the original vector. 7. This property is fundamental in understanding linear transformations and their geometric interpretations.