What is the defining quality of a transformation that is not invertible?

The defining quality of a transformation that is not invertible is that it changes the dimensionality of the input space. When a transformation alters the dimensionality, such as projecting a 3-dimensional object onto a 2-dimensional plane or mapping a vector from a higher-dimensional space to a lower-dimensional space (for example, from ( \mathbb{R}^2 ) to ( \mathbb{R}^1 )), it loses information, making it impossible to uniquely recover the original input from the output. This loss of information is the core reason why the transformation cannot be inverted.

In contrast, having an eigenvalue of 1 does not necessarily imply non-invertibility; such a transformation could still be invertible. A transformation can indeed have a non-zero determinant while still being non-invertible if it alters dimensionality. Similarly, preserving angles refers to properties of certain transformations (like isometries) but does not pertain to invertibility directly. Hence, changing the dimensionality is the clear characteristic that indicates a transformation cannot be inverted.