AI Math Mid-Term Summary - #1. Vector

hersheythings·2021년 10월 23일
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기초 수학 및 확률 및 통계 기본

기본적 수학 개념 : Vector


Scalar : single number

  • 0th0^{th} order tensor
  • example : 1,0.2,...1, -0.2, \,...

Vector : array of numbers

  • 1st1^{st} order tensor
  • example : v=[0.1,0.7,0.2]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 : [00]\begin{bmatrix} 0 \\ 0 \\ \end{bmatrix}, 굵은 글씨로 00 또는 0\overrightarrow{0}로 표시

Types of Vector

  • Unit Vector
    - 크기가 1
    - example : [1/21/2]\begin{bmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ \end{bmatrix}

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

  • Column Vector
    - v=[0.10.70.2]v = \begin{bmatrix} 0.1 \\ 0.7 \\ -0.2\\ \end{bmatrix}

Vector Operations

  • Addition
  • Scalar Multiplication
  • Dot Product
    - For v=[a1...an],u=[b1...bn],vu=i=1naibi\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)vu=vuTNote)\,\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 VV is a function, x:VR||\vec{x}|| : V \to \mathbb{R}
    • xxs.t.λ Randx,yV\vec{x}\to||\vec{x}||\,\,s.t.\,\,\forall\,\lambda\ \in \mathbb{R}\,\,and\,\,\vec{x},\vec{y}\in V the followings hold
      • Absolutely homogenous : λx=λx||\lambda \vec{x}|| =|\lambda|||\vec{x}||
      • Triangle inequality :x+yx+y||\vec{x}+\vec{y}|| \leq||\vec{x}||+||\vec{y}||
      • Positivie definite : x0andx=0||\vec{x}||\geq0\,\,and \,\,||\vec{x}||=0 if and only if x=0\vec{x} =0
  • example
    - The length of xinR2,whichimpliesL2norm\vec{x}\,in\,\mathbb{R}^{2}, which \,\,implies\,\,L_{2}\,norm
    • x2=(x12+x22)1/2=x12+x22||\vec{x}||_{2}=(x_{1}^2+x_{2}^2)^{1/2}=\sqrt{x_{1}^{2}+x_{2}^{2}}

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


Further Topics

Matrix : 2-D array

  • 2nd2^{nd} order tensor
  • example : A = [0.10.21.31.40.50.6]\begin{bmatrix} 0.1 & 0.2 \\ 1.3 & -1.4 \\ -0.5 & 0.6 \end{bmatrix}

Tensor

  • 만약 2차원보다 더 큰 차원을 표현하고 싶다면 ??
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