Relation, Function, Transformation, Mapping, Model, System의 정의를 명확히 파악하고, 여기서 linear transform 과 linearity 의 개념을 확실히 기억할 것.

Linear Transformations

<aside> 💡 A transformation (or function or mapping) $T$ from $\mathbb{R}^n$ to $\mathbb{R}^m$ is a rule that assigns each vector $\textbf{x}$ in $\mathbb{R}^n$ to a vector $T(\textbf{x})$ in $\mathbb{R}^m$.

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The set $\mathbb{R}^n$ is called domain (정의역) of $T$, and $\mathbb{R}^m$ is called codomain (공역) of $T$.
The notation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ indicates that the domain of $T$ is $\mathbb{R}^n$ and the codomain is $\mathbb{R}^m$.
For $\textbf{x} \in \mathbb{R}^n$, the vector $T(\textbf{x}) \in \mathbb{R}^m$ is called the image (상) of $\textbf{x}$ (under the action of $T$).

Figure 2

The set of all images $T(\textbf{x})$ is called the range (치역) of $T$. See Fig. 2 above

Matrix Transformations (=Linear transformation)

For each $\textbf{x} \in \mathbb{R}^n$, $T(\textbf{x})$ is computed as $A\textbf{x}$, where $A$ is an $m\times n$ matrix.
For simplicity, we denote such a matrix transformation by $\textbf{x} \mapsto A\textbf{x}$. *** 매우매우 중요.
Observe that the domain of $T$ is $\mathbb{R}^n$ when $A$ has $n$ columns and the codomain of $T$ is $\mathbb{R}^m$ when each column of $A$ has $m$ entries.
The range of $T$ is the set of all linear combinations of the columns of $A$, because each image $T(\textbf{x})$ is of the form $A\textbf{x}$.

Example 1:

Let $A=\begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix}, \textbf{u} =\begin{bmatrix} 2 \\ -1\end{bmatrix}, \textbf{b} =\begin{bmatrix} 3 \\ 2 \\ -5 \end{bmatrix}, \textbf{c} =\begin{bmatrix} 3 \\ 2 \\ 5\end{bmatrix}$ and

define a transformation $T: \mathbb{R}^2 \mapsto \mathbb{R}^3$ by $T(\textbf{x})=A\textbf{x}$,

so that $T(\textbf{x})=A\textbf{x}=\begin{bmatrix} 1 & -3 \\ 3 & 5 \\ -1 & 7 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =\begin{bmatrix} x_1-3x_2 \\ 3x_1 + 5x_2 \\ -1x_1 + 7x_2 \end{bmatrix}$.

Find $T(\textbf{u})$, the image of $\textbf{u}$ under the transformation $T$.
Find $\textbf{x} \in \mathbb{R}^2$ whose image under $T$ is $\textbf{b}$.
Is there more than one $\textbf{x}$ whose image under $T$ is $\textbf{b}$?
Determine if $\textbf{c}$ is in the range of the transformation $T$.

Solution of Example 1