Kernel and Range

A rigorous treatment of null spaces, images, bases, dimensions, and the rank–nullity theorem.

Vector-space ideas connect sets, operations, spanning, independence, basis, and dimension. A vector space sits at the centre of a dependency map. Subspace conditions lead to span and independence, which meet at basis and determine dimension; kernel and range connect through a transformation.

Precise definition

For a linear map $T:V\to W$, the kernel is $\ker T=\{v\in V:T(v)=0\}$ and the range or image is $\operatorname{range}T=\{T(v):v\in V\}\subseteq W$. Both are subspaces: the kernel of the domain and range of the codomain.

Notation and mathematical language

For a matrix map $T(x)=Ax$, $\ker T$ is the null space of $A$, while $\operatorname{range}T$ is the column space of $A$. Nullity is $\dim\ker T$ and rank is $\dim\operatorname{range}T$. Rank–nullity gives $\dim V=\operatorname{nullity}T+\operatorname{rank}T$.

Conceptual picture

The kernel measures inputs erased by the map, so $T$ is injective exactly when $\ker T=\{0\}$. The range records reachable outputs, so $T$ is surjective exactly when its range equals the codomain.

Fully worked example

Interpretation and application

Kernel and range describe information loss, attainable outputs, identifiability, and constraints. A large kernel means many inputs produce the same output; that can be a designed compression or an obstacle to reconstruction.