> [!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
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