[[Linear algebra MOC]]
# Linear isomorphism

A  $\mathbb{K}$-**linear isomorphism** $\varphi : V \to W$ is an isomorphism in [[Category of vector spaces]], #m/def/linalg 
i.e. a [[linear map]] with a (necessarily unique) two-sided inverse $\varphi^{-1}: W \to V$
such that $\varphi^{-1}\varphi=1_{V}$ and  $\varphi\varphi^{-1} = 1_{W}$.

## Properties

- A linear map $\varphi \in \Vect_{\mathbb{K}}(V,W)$ is an isomorphism iff it is a [[linear monomorphism]] and [[Split epimorphism|split]] [[linear epimorphism]].
- See [[Rank-nullity theorem#Corollaries]].

#
---
#state/tidy | #lang/en | #SemBr