[[Linear map]]
# Linear epimorphism

Let $T : U \to V$ be a $\mathbb{K}$-[[linear map]].
The following statements are equivalent: #m/thm/linalg 

- $T$ is [[Surjectivity, injectivity, and bijectivity|surjective]];
- $T$ is [[Morphism|epic]] in [[Category of vector spaces]];
- $T$ is [[Split epimorphism|split epic]] in [[Category of vector spaces]] (assuming [[Axiom of Choice|AC]])l
- $\coker T = \{ 0 \}$.

> [!check]- Proof
> See [[Module epimorphism]]

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