> [!summary] Matrix mapping is an analytical representation to describe linear mapping for computational purposes > **Common Notation:** > **Transformation Notation:** $T (\vec{x}) = A \vec{x}, \quad \vec{x} \in \mathbb{R}^n$ > **Linear Mapping Notation:** $L(\vec{x}) = A \vec{x}, \quad \vec{x} \in \mathbb{R}^n$ >[!info]+ Read Time **⏱ 1 min** # Definition Matrix mapping is an analytical representation of [[Linear Mapping (Transformations)|linear mapping]]. It transforms a [[Scalar & Vectors|vector]] using [[Matrix-Vector Multiplication|matrix-vector multiplication]] from its [[Domain & Codomain|domain]] in $\mathbb{R}^n$ to its [[Domain & Codomain|codomain]] in $\mathbb{R}^m$. Matrix mapping is usually denoted as a function ($f_{A}$), but can also be denoted using transformation notation ($T$), or linear transformation notation ($L$), described below $ \begin{array}{c} \text{Function notation} \\ f_{A}(\vec{x}) = A \vec{x}, \quad \vec{x} \in \mathbb{R}^n \\ \\ \text{Transformation notation} \\ T(\vec{x}) = A \vec{x}, \quad \vec{x} \in \mathbb{R}^n \\ \\ \text{Linear notation} \\ L(\vec{x}) = A \vec{x}, \quad \vec{x} \in \mathbb{R}^n \\ \end{array} $ [!note] Matrix $A$ is sometimes called the standard matrix, which is how the standard basis changes as a result to the transformation # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/DRHJh4wrLkY?si=E-A0aO9MAUQmgZBr" 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>