Projection onto the ZAxis

Given a transformation that projects the space onto the z-axis, where did the vectors in $\mathbb{R}^3$ go, and are there any vectors that were mapped to the zero vector?

The problem of understanding how a transformation projects the space of three-dimensional vectors onto the z-axis provides an insightful look into linear maps and their effects on vector spaces. When you project a vector space onto one specific axis, you are essentially dropping the components perpendicular to that axis, leaving you with vectors that only have components along the z-axis. Conceptually, this means that every point from the three-dimensional space is 'flattened' onto the z-axis, focusing on the z-component of each vector while the x and y components become zero. This process highlights the role of projections in simplifying vector representations and conserving direction along a specified axis.

In terms of linear algebra, vectors that initially lie in the xy-plane will be mapped to the zero vector since their z-component was zero from the start. This concept is crucial as it touches on the idea of the kernel of a linear transformation, or the null space, which consists of all vectors that get mapped to the zero vector. For this z-axis projection, the null space is the entire xy-plane, showcasing a practical example of understanding a transformation's kernel. Additionally, identifying vectors that remain unchanged allows one to understand the concept of fixed points of a transformation, deepening comprehension of vector space manipulation. This completely underlines the importance of visualizing transformations in theoretical and practical aspects of vector calculus.