Transformation and Line Mapping in R2

For the transformation, identify all vectors that fall on the line $y = -2x$ and are transformed to the zero vector, and describe the image set which consists of vectors that land on the line $y = x$.

This problem focuses on understanding the effects of transformation on vectors, particularly identifying vectors that transform to the zero vector and describing image sets. The specific line given, $y = -2x$, suggests that the transformation involves a kind of geometric interpretation where vectors align in a specific manner with this line. To find vectors transformed to the zero vector, you need to consider the kernel (or null space) of the transformation, where a nontrivial solution might stretch or shrink vectors to nullify their contributions yielding the zero vector.

Moreover, examining the image set which lands on the line $y = x$ provides an opportunity to delve into the analysis of transformation images. This involves identifying how vectors are mapped by the transformation and how the linear map affects their resultant direction and magnitude. Such problems help illuminate the nature of vector spaces and the impact of linear maps on these spaces. They serve as a foundation for more advanced studies in transformations and vector mapping, increasing one's understanding of concepts like null space, image, and rank, which are crucial in further exploration of linear algebraic structures.