64. Preimage
For a transformation and a subset , the preimage of under is the set of inputs that map into :
For a single point , the preimage may be empty, a single point, or an entire affine subspace — depending on .
Example
Let be defined by , so .
Preimage of :
Solve :
Preimage of : similarly .
Preimage of the set :
64.1. Properties
- For a linear : is exactly the kernel of
- For linear and any : for any particular solution — an affine subspace parallel to the kernel
64.2. Connections
- Image — the forward direction
- Image of a subset — applying to a set rather than
- Kernel — preimage of the zero vector under a linear map