> [!summary] Linear mapping is a function that transforms vectors between vector spaces. > **Properties:** > 1. $L(\vec{u}+\vec{v})=L(\vec{u})+L(\vec{v})$ (Vector addition) > 2. $cL(\vec{v}) = L(c\vec{v})$ (Scalar multiplication ) >[!info]+ Read Time **⏱ 1 min** # Definition Linear mapping is a function that transforms [[Vectors Addition & Multiplication|vectors]] between [[Vector Spaces|vector spaces]]. Meaning that linear mapping [[Domain & Codomain|maps]] a [[Vector Spaces|vector space]] ($V$) to another [[Vector Spaces|vector space]] ($W$). > [!note]+ Linear Transformation Diagram > ![[lm_1.png]] > Linear mapping a vector from a vector space $V$ to $W$ ## Conditions for Linear Mapping > [!warning] Assumptions Mathematically, to linearly map ($L$) a vector ($\vec{x}$) from a vector space to a subspace, it must abide by the conditions of a subspace. So assume the following: > - All vectors $\vec{u},\vec{v}$ are in the [[Subspace|subspace]] ($\vec{u},\vec{v}\in W$) > - $c$ represents a [[Field|field of scalars]] ($c\in F$) So linear mapping must satisfy these two properties. 1. $L(\vec{u}+\vec{v})=L(\vec{u})+L(\vec{v})$ (Vector addition) 2. $cL(\vec{v}) = L(c\vec{v})$ (Scalar multiplication ) 3. $L(\vec{0})=L(\vec{0})$ Zero vector If vector addition and scalar multiplication are satisfied, then the two properties are called **preserved under additivity** (vector addition) and **homogeneity** (scalar multiplication), # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/kYB8IZa5AuE?si=P6Vj4CAT-T6YwiS4" 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>