인공지능을 위한 선형대수 - CHAPTER 2.5 선형변환

www.edwith.org/linearalgebra4ai/lecture/23826/

[LECTURE] 선형변환 : edwith

Transformation

A transformation, function or mapping, \(T\) maps an input \(x\) to an output \(y\)

Mathematical notation: \(T: x \rightarrow y\)

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

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

Example:

$$T: \mathbf{x} \in \mathbb{R}^{3} \rightarrow \mathbf{y} \in \mathbb{R}^{2} ,\ \mathbf{x}=\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \in \mathbb{R}^{3} \  \rightarrow \mathbf{y}=T(\mathbf{x})=\begin{bmatrix} 4 \\ 5 \end{bmatrix} \in \mathbb{R}^{2}$$

Matrix of Linear Transformation

Example:

Suppose \(T\) is a linear transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{3}\) such that \(T(\begin{bmatrix} 1 \\ 0 \end{bmatrix}) = \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}\) and \(T(\begin{bmatrix} 0 \\ 1 \end{bmatrix}) = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}\).

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

 

$$\mathbf{x}= \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = x_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + x_{2}\begin{bmatrix} 0 \\ 1 \end{bmatrix}$$  
$$\rightarrow T(\mathbf{x})=T(x_{1}\begin{bmatrix} 1 \\ 0 \end{bmatrix} + x_{2}\begin{bmatrix} 0 \\ 1 \end{bmatrix}) = x_{1}T(\begin{bmatrix} 1 \\ 0 \end{bmatrix}) + x_{2}T(\begin{bmatrix} 0 \\ 1 \end{bmatrix}) = x_{1}\begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix} + x_{2}\begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}=\begin{bmatrix} 2 && 0 \\ -1 && 1 \\ 1 && 2 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}$$

여기서, \(\begin{bmatrix} 2 && 0 \\ -1 && 1 \\ 1 && 2 \end{bmatrix}\)를 Standard Matrix라 칭합니다.

 

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

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

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 \(\mathbb{R}^{n \times n}\):

$$A = \begin{bmatrix} T(\mathbf{e}_{1}) && \cdots && T(\mathbf{e}_{n}) \end{bmatrix}$$

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.

$$T(\mathbf{x}) = x_1T(\mathbf{e}_{1}) + \cdots + x_nT(\mathbf{e}_{n})$$

 

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