Kernel, Image, and Rank–Nullity

Kernel and Image

Two subspaces describe any linear map $T : V \to W$. The kernel ($\ker T$) is everything $T$ sends to zero: $\ker T = \{v \in V : T(v) = \mathbf{0}\}$. It measures what the map collapses. $T$ is injective exactly when $\ker T = \{\mathbf{0}\}$.

The image ($\operatorname{im} T$) is everything $T$ can produce: $\operatorname{im} T = \{T(v) : v \in V\}$. It measures what the map reaches. The dimension of the image is the rank, and the dimension of the kernel is the nullity.

Rank–Nullity Theorem

The Rank–Nullity Theorem links these dimensions to the domain: $$\dim(\ker T) + \dim(\operatorname{im} T) = \dim V$$

Nullity + rank = dimension of the domain. Every dimension of the input is accounted for: some collapse (kernel) and the rest survive (image). If a map from $\mathbb{R}^5$ has a 2D kernel, its image must be 3D.

Pitfall: Confusing the kernel (in the domain) with the image (in the codomain), or treating Rank–Nullity as optional. It is a hard accounting constraint: one forces the other.