61. Surjective, Injective, Bijective
Three properties a function might (or might not) have, with clean characterizations for linear transformations.
61.1. Injective (one-to-one)
Different inputs produce different outputs — no two distinct inputs collide:
For a linear , equivalently:
(See Kernel — only the zero vector maps to zero.)
61.2. Surjective (onto)
Every element of the codomain is hit by some input:
For a linear , equivalently:
61.3. Bijective (one-to-one and onto)
Both injective and surjective. Each output has exactly one preimage. Bijective linear transformations are invertible.
For a linear , bijective requires (square matrix) and is invertible.
61.4. Summary table for linear given by matrix
| Property | Equivalent (rank) | Equivalent (kernel/image) | Possible only if |
|---|---|---|---|
| Injective | (full column rank) | ||
| Surjective | (full row rank) | ||
| Bijective | both above | and invertible |
Example
— .
- → injective
- → not surjective
- Not bijective ()
61.5. Connection to inverse
A linear has an inverse iff it’s bijective iff is square and invertible.
When invertible, corresponds to (see Matrix Inverse).