62. Image
The image (or range) of a linear transformation is the set of all possible outputs:
This is a generalization of the image of a subset applied to the entire domain.
62.1. Matrix representation
If is represented by an matrix with columns , then for any input :
— a linear combination of ‘s columns.
62.2. Image = Column space
The image of equals the column space of :
Example
, .
(The two columns are linearly independent, so they span .)
62.3. Properties
- is a subspace of the codomain
- rank of the matrix
- Rank–Nullity:
- is surjective iff iff has full row rank
62.4. Three names for the same thing
- Image of — emphasis on the linear-map perspective
- Range of — same, common in analysis
- Column space of — same, matrix-algebra perspective