인공지능을 위한 선형대수 - CHAPTER 2.5 선형변환

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[LECTURE] 선형변환 : edwith

Transformation

A transformation, function or mapping, $T$ maps an input $x$ to an output $y$

Mathematical notation: $T:x\to y$

Domain(정의역): Set of all the possible values of $x$

Co-domain(공역): Set of all the possible values of $y$

Image(상): a mapped output $y$, given $x$

Range(치역): Set of all the output values mapped by each $x$ in the domain

 

Note: the output mapped by a particular $x$ is uniquely determined.

Linear Transformation

A transformation (or mapping) $T$ is linear if:

$$T(c\mathbf{u}+d\mathbf{v})=cT(\mathbf{u})+dT(\mathbf{v})$$

for all $\mathbf{u},\mathbf{v}$ in the domain of $T$ and for all scalars $c$ and $d$

Transformations between Vectors

$T:\mathbf{x}\in {\mathbb{R}}^n\to \mathbf{y}\in {\mathbb{R}}^m$: Mapping $n$-dim vector to $m$-dim vector

Example:

$$T:\mathbf{x}\in {\mathbb{R}}^3\to \mathbf{y}\in {\mathbb{R}}^2,\mathbf{x}=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\in {\mathbb{R}}^3\to \mathbf{y}=T(\mathbf{x})=\left[\begin{array}{c}4\\ 5\end{array}\right]\in {\mathbb{R}}^2$$

Matrix of Linear Transformation

Example:

Suppose $T$ is a linear transformation from ${\mathbb{R}}^2$ to ${\mathbb{R}}^3$ such that $T(\left[\begin{array}{c}1\\ 0\end{array}\right])=\left[\begin{array}{c}2\\ -1\\ 1\end{array}\right]$ and $T(\left[\begin{array}{c}0\\ 1\end{array}\right])=\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right]$.

With no additional information, find a formula for the image of an arbitrary $\mathbf{x}$ in ${\mathbb{R}}^2$

 

$$\mathbf{x}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=x_1\left[\begin{array}{c}1\\ 0\end{array}\right]+x_2\left[\begin{array}{c}0\\ 1\end{array}\right]$$  
$$\to T(\mathbf{x})=T(x_1\left[\begin{array}{c}1\\ 0\end{array}\right]+x_2\left[\begin{array}{c}0\\ 1\end{array}\right])=x_{1}T(\left[\begin{array}{c}1\\ 0\end{array}\right])+x_{2}T(\left[\begin{array}{c}0\\ 1\end{array}\right])=x_1\left[\begin{array}{c}2\\ -1\\ 1\end{array}\right]+x_2\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right]=\left[\begin{array}{ccc}2& & 0\\ -1& & 1\\ 1& & 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$$

여기서, $\left[\begin{array}{ccc}2& & 0\\ -1& & 1\\ 1& & 2\end{array}\right]$를 Standard Matrix라 칭합니다.

 

In general, let $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ be a linear transformation. Then $T$ is always written as a matrix-vector multiplication, i.e.,

$$T(\mathbf{x})=A\mathbf{x}\mathsf{for}\mathsf{all}\mathbf{x}\in {\mathbb{R}}^n$$

In fact, the $j$-th column of $A\in {\mathbb{R}}^{m\times n}$ is equal to the vector $T({\mathbf{e}}_j$, where ${\mathbf{e}}_j$ is the $j$-th column of the identity matrix in ${\mathbb{R}}^{n\times n}$:

$$A=\left[\begin{array}{ccccc}T({\mathbf{e}}_1)& & \cdots & & T({\mathbf{e}}_n)\end{array}\right]$$

Here, the matrix $A$ is called the standard matrix of the linear transformation $T$.

 

Linear Transformation $T$의 입력 벡터 $\mathbf{x}$를 표준 기저(Standard Basis) 벡터에 대한 Linear Combination으로도 표현할 수 있습니다.

$$\mathbf{x}=x_1{\mathbf{e}}_1+\cdots +x_n{\mathbf{e}}_n$$

그리고, Linear Transformation의 특징에 따라 Linear Transformation $T$를 아래와 같이 표현할 수 있습니다.

$$T(\mathbf{x})=x_{1}T({\mathbf{e}}_1)+\cdots +x_{n}T({\mathbf{e}}_n)$$

 

어떤 변환이 Linear Transformation이라면, 이를 행렬과 입력 벡터간의 곱으로 나타낼 수 있습니다. 즉, 행렬 곱은 Linear Transoformation으로 정의할 수 있습니다.