> [!summary] The composition of linear mappings is a mapping of multiple vector spaces in order. > **Key Equations:** $(T_{2}\circ T_{1})(x)=T_{2}(T_{2}(\vec{x}))$ >[!info]+ Read Time **⏱ 1 min** # Definition If a vector ($\vec{x}$) is [[Linear Mapping (Transformations)|linearly mapped]] from vector spaces $V,W,P$ via [[Linear Mapping (Transformations)|transformation]] $L_{1}: V\to W$ & $L_{2}: W\to P$. Then the composition of $T_{2}$ with $T_{1}$ denoted as $T_{2}\circ T_{1}$ is the composition of linear transformation formally defined below. $ \begin{array}{c} (T_{2}\circ T_{1})(x)=T_{2}(T_{2}(\vec{x})) \\ \\ \text{Where} \\ (T_{2}\circ T_{1}) : V \to P \\ \end{array} $ > [!note]+ Composition of Linear Mapping Diagram ![[colm_1.png]] Since $T_{1}$ maps $\vec{x}$ from the vector space $V$ to $W$, and $T_{2}$ maps $\vec{x}$ from vector space $W$ to $P$. Then the composition of $T_{2},T_{1}$ is the shortcut from start to end.