> [!summary] The null space is a set of vectors that are mapped to the zero vector. > **Key Equations:** $\text{Null}(L)=\left\{ A \vec{x}=\vec{0} \quad |\vec{x} \in \mathbb{R}^n \right\}$ >[!info]+ Read Time **⏱ 2 mins** # Definition The null space is an analytical representation of a [[Kernel|kernel]]. The null space of a [[Matrix Mapping (Transformations)|linear transformation]] ($L: \mathbb{R}^n\to \mathbb{R}^m$) is a [[Subspace|subspace]] of the [[Domain & Codomain|domain]] $\mathbb{R}^n$, that are mapped to the [[Zero Vector|zero vector]] in the [[Domain & Codomain|codomain]] $\mathbb{R}^m$. The null space is written as $\text{Null}(L)$ formally defined below $ \begin{array}{c} \text{Null}(L) &= \left\{ L(\vec{x})=\vec{0} \quad |\vec{x} \in \mathbb{R}^n \right\} \\ &= \left\{ A \vec{x}=\vec{0} \quad |\vec{x} \in \mathbb{R}^n \right\} \end{array} $ > [!note]+ Visualization of a null space ![[np_g_1.gif]] The linear transformation of a set of vectors from $\mathbb{R}^2\to \mathbb{R}^1$ mapping to the zero vector. ## Proof of Subspace > [!warning] Assumptions To prove null space is a [[Subspace|subspace]] of the domain through the [[Subspace|subspace tests]], assume the following: > - If $\vec{x},\vec{y}\in \text{null}(A)$ then $A\vec{x} = \vec{0}$ and $A\vec{x}=\vec{0}$ > - $c\in \mathbb{R}$ **Zero vector:** $A \vec{0}= \vec{0} \quad \Rightarrow \vec{0}\in \text{null}(A)$ **Closed under vector addition:** $ A(\vec{x} +\vec{y})=A \vec{x}+A \vec{y}= \vec{0}+ \vec{0}=\vec{0} \quad \Rightarrow \vec{x} + \vec{y} \in \text{null}(A) $ **Closed under scalar multiplication:** $ A(c \vec{x} ) = c(A \vec{x}) =c \cdot \vec{0} = \vec{0} \quad \Rightarrow c \vec{x} \in \text{null}(A) $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/YQRioQ1XUck?si=48MS6nRgxx-0VBdK" 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> --- > ✍️ This project’s been a labour of love. > If it helped, [give Math & Matter a star](https://github.com/rajeevphysics/Obsidian-MathMatter) and let me know what you'd like to see next. ---