> [!NOTE] Definition (Image) > 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 $W$ $\text{Im}(\varphi) = \{ \varphi(v) \mid v \in V \}$ > [!NOTE] Definition (Kernel) > 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 $\varphi$)* . # Properties > [!NOTE] Proposition (Kernel and Image are subspaces) > Suppose $\varphi:V\to W$ is a [[Linear maps|linear map]]. Then > 1. $\ker \varphi$ is a [[Vector subspace|subspace]] of $V$; > 2. $\text{Im}\, \varphi$ is a *subspace* of $W$. ^4f5b10 >*Proof* >1. We know that $0_{V}\in \ker \varphi$ since $\varphi(0_{V})=0_{W}$ ([[Linear maps#^7ce00d|linear maps fix the zero vector]]). Suppose $v_{1},v_{2}\in \ker \varphi$ and $\lambda_{1},\lambda_{2}\in \mathbb{F}$. Then by linearity of $\varphi$ $\varphi (\lambda_{1} v_{1} + \lambda_{2} v_{2}) = \lambda_{1} \varphi(v_{1}) + \lambda_{2} \varphi(v_{2}) = \lambda_{1} 0_{V} + \lambda_{2} 0_{V} = 0_{V}$and so $\lambda_{1}v_{1}+\lambda_{2}v_{2} \in \ker \varphi$, as required. >2. > >Naturally we are interested in the dimension of these subspaces (in the case where $V$ is finite dimensional). > [!NOTE] Corollaty (If $V$ is finite dimensional then so are the kernel and image of $\varphi$) > Contents ^616c3c >*Proof.* >ker is a subspace so is finite dimensional by [[Vector subspace#^dc4bab|dimension of subspace of FDVS]]. >Im is finite dimensional by [[Linear maps#^a8ff22|linear maps are equal if they agree on basis elements]]. > > [!NOTE] Theorem (Rank-Nullity Formula) > Suppose $V$ is [[Vector spaces|finite dimensional]]. For any linear map $\varphi :V \to W$, $\dim \text{Im } \varphi + \dim \ker \varphi = \dim V$ > > >See [[Rank-nullity formula|proof]].