[[Linear map]]
# Kernel of a linear map

The **kernel** $\ker T$ or **null space** of a [[linear map]] $T \in \Vect_{\mathbb{K}}(U,V)$ is the the [[Image and preïmage|preïmage]] $T^{-1} \{ \vab 0 \}$, #m/def/linalg 
i.e. the set of all vectors in $U$ that are mapped to $\vab 0$.
It is therefore equivalent to the [[Kernel of a group homomorphism]] of $T$ considered as a [[group homomorphism]].
The **nullity** $\nullity T$ of a linear map is the [[Dimension of a vector space|dimension]] of its kernel. #m/def/linalg 

## Properties

- [[Rank-nullity theorem]]
- If $\vab u$ is a solution to $A \vab v = \vab 0$ then the full solution set is $\vab u + \ker A$

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