> [!summary] Range is the image of an entire vector space. > **Key Equations:** $\text{Range}(L) = \mathrm{Im}(L) = \left\{ L (\vec{v}) \space | \vec{v} \in V \right\}$ >[!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 $L: V\to W$. Range is the [[Image|image]] of the entire [[Vector Spaces|vector space]] $V$. So the range would be a [[Subspace|subspace]] of $W$ formally defined below. $ \text{Range}(L) = \mathrm{Im}(L) = \left\{ L(\vec{v}) \space | \vec{v} \in V \right\} $ ![[ra_1.png]] ## Proof of Subspace > [!warning] Assumptions To prove that $\text{Range(A)}\subseteq W$ through [[Direct Proof|direct proof]], assume the following: > - $\exists v_{1},v_{2} \in V$ so that $L(v_{1})=w_{1} \text{ and }L(v_{2})=w_{2} \quad | w_{1,}w_{2}\in W$ > - The [[Zero Vector|zero vector]] is denoted as $\vec{0}$ > - $c$ is a [[Field|field of scalars ]] **Zero Vector:** $ L(\vec{0}) = \vec{0} \in W, \quad \vec{0}\in \text{Range}(L) $ **Vector Addition:** $ w_{1}+w_{2}= L(v_{1}) + L(v_{2}) = L(v_{1}+v_{2})\in W ,\quad L(v_{1}+v_{2})\in \text{Range(L)} $ **Scalar Multiplication:** $ cw_{1} = cL(v_{1}) = L(cv_{1}) \in W, \quad L(cv_{1})\in \text{Range}(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>