Linear Map

A linear map $T : V \to W$ is a function between vector spaces over the same field that preserves addition and scalar multiplication: $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(a\mathbf{v}) = aT(\mathbf{v})$.

Linear maps are the structure-preserving maps (homomorphisms) of vector spaces. The kernel $\ker(T) = \{\mathbf{v} \mid T(\mathbf{v}) = \mathbf{0}\}$ and image $\operatorname{im}(T) = \{T(\mathbf{v}) \mid \mathbf{v} \in V\}$ are subspaces, and the rank-nullity theorem states $\dim V = \dim \ker(T) + \dim \operatorname{im}(T)$.

In category theory, linear maps are the morphisms in the category of vector spaces over a field.