65. Kernel
For a linear transformation , the kernel (or null space of ) is the set of vectors that map to the zero vector:
The kernel is the preimage of .
65.1. Why the kernel is a subspace
For any and scalar :
- → closed under addition
- → closed under scalar multiplication
- since
Hence is a subspace of .
65.2. Matrix form: kernel ↔ null space
If is represented by a matrix (so ), the kernel of equals the null space of :
Example
Let with . Solve :
Only the trivial solution: . The transformation is injective.
65.3. Injectivity
A linear transformation is injective (one-to-one) if and only if .
Proof sketch: if , then , so . The kernel being trivial forces .
65.4. Connections
- Null Space — kernel of the matrix representation
- Rank–Nullity:
- Image — the “output side” counterpart
- Preimage — generalization to non-zero target sets
- Zero Vector — what kernel elements map to