> [!NOTE] Definition (Image & Kernel) > Let $\varphi:V \to W$ be a [[Linear maps|linear map]]. The [[Image of a set under a function|image]] of $\varphi$, denoted $\text{Im}(\varphi)$, is the subset of $W$ $\text{Im}(\varphi) = \{ \varphi(v) \mid v \in V \}$ > >The **kernel** of $\varphi$, denoted $\ker (\varphi)$, is [[Preimage (of set under a function)|pre-image]] of $0_{W}$ under $\varphi$: $\ker(\varphi) = \{ v \in V \mid \varphi(v) = 0_{W} \}$ *(i.e the set of roots of $\varphi$)* .