> [!summary] The kernel is the set of vectors in the domain of a transformation, which, when transformed, are mapped to the zero vector. > **Key Equations:** $\text{ker}(L) = \left\{L (\vec{v})=\vec{0} | \space \vec{v} \in V \right\}$ >[!info]+ Read Time **⏱ 1 min** # Definition If a [[Vector Spaces|vector space]] $V$ was [[Linear Mapping (Transformations)|linearly mapped]] onto another [[Vector Spaces|vector space]] $W$ via transformation $L: V\to W$. The kernel of $L$ written as $\ker(L)$ is the [[Subspace|subspace]] in $V$ that are [[Linear Mapping (Transformations)|mapped]] to the [[Zero Vector|zero vector]] in $W$: $ \ker(L) = \left\{L(\vec{v})=\vec{0} | \space \vec{v} \in V \right\} $ ![[ker_1.png]] ## Proof of Subspace > [!warning] Assumptions To prove kernel is a [[Subspace|subspace]] through the [[Subspace|subspace test]], assume the following: > - The [[Zero Vector|zero vector]] is defined as $\vec{0}\in V$ > - $\exists \vec{u},\vec{v}\in V$ and $L(\vec{u})=0\in W$ and $L(\vec{v})\in W$ > - $\alpha \in \mathbb{R}$ **Zero vector:**   $ L(\vec{0}) = \vec{0}, \quad \vec{0}\in \ker{L} $ **Closed under addition:**   $ L(\vec{u}+\vec{v}) = L(\vec{u}) + L(\vec{v}) = 0 + 0 = 0 \quad\Rightarrow\quad \vec{u}+\vec{v} \in \ker(L) . $ **Closed under scalar multiplication:**   $ L(\alpha \vec{v}) = \alpha L(\vec{v}) = \alpha \cdot 0 = 0 \quad\Rightarrow\quad \alpha \vec{v} \in \ker(L) . $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/vyYrvhbDhW4?si=9p72z9REUj1u0iuC" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>