📂Matrix Algebra

Matrix Transformation

Definition

A function from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ is called a matrix transformation with respect to the matrix $m \times n$ $A$ if it maps as follows, and is denoted as $T_{A} : \mathbb{R}^{n} \to \mathbb{R}^{m}$.

$$ \mathbf{w} = T_{A} (\mathbf{x}) = A\mathbf{x}\quad \left( \mathbf{x} \in \mathbb{R}^{n}, \mathbf{w} \in \mathbb{R}^{m} \right) $$

It can also be represented as $\mathbf{x} \overset{T_{A}}{\to} \mathbf{w}$.

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This mapping can be represented in matrix form as follows.

$$ \begin{bmatrix} w_{1} \\ w_{2} \\ \vdots \\ w_{m} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix} $$

When represented in the form of a linear system, it is as follows.

$$ \begin{align*} w_{1} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \\ w_{2} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} \\ &\vdots \\ w_{m} &= a_{m1}x_{1} + a_{m2}x_{2} + \cdots + a_{mn}x_{n} \end{align*} $$

Description

Matrix transformation is a type of linear transformation.