AI Math Mid-Term Summary - #1. Vector

기초 수학 및 확률 및 통계 기본

기본적 수학 개념 : Vector

Scalar : single number

• $0^{th}$ order tensor
• example : $1, -0.2, \,...$

Vector : array of numbers

• $1^{st}$ order tensor
• example : $v=[0.1, 0.7, -0.2]$
• 속도나 물리적인 힘처럼 "방향"이 있는 것
• Zero Vector (null vector) : vector의 시점과 종점이 동일
  - 크기가 0, addcitive identity(항등원) in a vector space
  - example : $\begin{bmatrix} 0 \\ 0 \\ \end{bmatrix}$, 굵은 글씨로 $0$ 또는 $\overrightarrow{0}$로 표시

Types of Vector

• Unit Vector
  - 크기가 1
  - example : $\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ \end{bmatrix}$

• Row Vector
  - $v = \begin{bmatrix} 0.1, 0.7, -0.2\\ \end{bmatrix}$

• Column Vector
  - $v = \begin{bmatrix} 0.1 \\ 0.7 \\ -0.2\\ \end{bmatrix}$

Vector Operations

• Addition
• Scalar Multiplication
• Dot Product
  - For $\vec{v} = [a_{1}...a_{n}], \vec{u}=[b_{1}...b_{n}],\, \vec{v}\cdot\vec{u}=\sum_{i=1}^na_{i}b_{i}$ where the size of each vector is equal.
  

$Note)\,\vec{v}\cdot\vec{u} =vu^{T}$

Norm

• Motivation

  어떻게 하면 주어진 벡터의 길이 또는 크기를 정의(측정)할 수 있을까?

• Definition
  - A norm on a vector space $V$ is a function, $||\vec{x}|| : V \to \mathbb{R}$
  • $\vec{x}\to||\vec{x}||\,\,s.t.\,\,\forall\,\lambda\ \in \mathbb{R}\,\,and\,\,\vec{x},\vec{y}\in V$ the followings hold
    • Absolutely homogenous : $||\lambda \vec{x}|| =|\lambda|||\vec{x}||$
    • Triangle inequality :$||\vec{x}+\vec{y}|| \leq||\vec{x}||+||\vec{y}||$
    • Positivie definite : $||\vec{x}||\geq0\,\,and \,\,||\vec{x}||=0$ if and only if $\vec{x} =0$
• example
  - The length of $\vec{x}\,in\,\mathbb{R}^{2}, which \,\,implies\,\,L_{2}\,norm$
  • $||\vec{x}||_{2}=(x_{1}^2+x_{2}^2)^{1/2}=\sqrt{x_{1}^{2}+x_{2}^{2}}$

$Note) L_{p}\, Norm :=(\sum_{i=1}^n|x_{i}|^{p})^{1/p}$

Further Topics

Matrix : 2-D array

• $2^{nd}$ order tensor
• example : A = $\begin{bmatrix} 0.1 & 0.2 \\ 1.3 & -1.4 \\ -0.5 & 0.6 \end{bmatrix}$

Tensor

• 만약 2차원보다 더 큰 차원을 표현하고 싶다면 ??