선형대수 Chapter 3.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " ## Chapter 3 Orthogonality"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### orthogonal vectors & Subspace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "왜 수직인 verctor들을 찾아야하나?\n",
    "\n",
    "- 독립적인 basis vector이기 때문이다.\n",
    "\n",
    "- 선형 결합으로 이루어져 있기때문에 계산하기 쉽다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### length of vectors ||x||"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "벡터의 길이는 다음과 같다. \n",
    "\n",
    "$$||x||^2 =  X^TX = \\sum_{i=1}^n x_i^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위 공식과 피타고라스 정리로 수직인 두 벡터의 내적의 값을 증명할 수 있다."
   ]
  },
  {
   "attachments": {
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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위와 같이 벡터 x, y, y-x로 이루어진 직각 삼각형이 있을때, 피타고라스 공식을 쓰면,\n",
    "\n",
    "$$ ||y-x||^2 = ||x||^2 + ||y||^2 $$\n",
    "\n",
    "$$ (y-x)^T(y-x) = x^Tx+y^Ty $$\n",
    "\n",
    "$$ y^Ty - y^Tx - x^Ty + x^Tx = x^Tx + y^Ty $$\n",
    "\n",
    "$$x^Ty = y^Tx,  x^Ty=0 $$\n",
    "\n",
    "이와 같이 벡터 x와 y가 수직일 때 벡터 x,y의 내적의 값은 0이다"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $x^Ty$의 내적의 특징"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- $x^Ty$ = 0 -> angle = 90도\n",
    "\n",
    "- $x^Ty$ < 0 -> angle > 90도\n",
    "\n",
    "- $x^Ty$ > 0 -> angle < 90도"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 0이 아닌 벡터 $V_1,V_2,...,V_n$이 모두 수직 일 때"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$V_i^TV_j =0 (i\\neq j) \\\\ ||V_j|| \\neq 0$ \n",
    "\n",
    "위 두 조건을 만족하면,\n",
    "\n",
    "$C_1V_1 + C_2V_2 + ... + C_n V_n = 0 \\\\ C_1 = C_2 = ... = C_n = 0$\n",
    "\n",
    "이기 때문에 벡터 $V$들은 서로 선형독립이다.\n",
    "\n",
    "따라서 $V_i^T(C_1V_1 + C_2V_2 + ... + C_n V_n)=0$ 이고\n",
    "\n",
    "$C_i||V_i||^2 = 0 for \\forall i \\\\ 즉, C_i = 0 이다.$\n",
    "\n",
    "따라서 orthogonal vector들은  벡터공간을 span 하는 basis vector들 이고,\n",
    "\n",
    "$X = \\sum_{i=1}^n C_i V_i = V_i^TX =C_i||V_i||^2$ 를 만족하기 때문에\n",
    "\n",
    "orthonormal $||V_j||=1$ 이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Orthogonal Subspaces"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Subspace에 있는 하나의 벡터가 다른 Subspace에서의 벡터와 수직이면 나머지 모든 벡터도 수직이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Row Space $\\perp$ Null space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Row Space $C(A^T)$와 Null space $X$는 언제나 서로 수직이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Column Space $\\perp$ Left null space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Column Space $C(A)$와 Left Null space $y$는 언제나 서로 수직이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### vector space를 span하는 독립 벡터들의 개수(vector space의 차원의 수)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "vector space $A_{m\\times n}$의 rank는 다음과 같다.\n",
    "\n",
    "- Dim(C(A)) + Dim(N($A^T$)) = m\n",
    "\n",
    "A의 column space의 차원수가 r 이고, $A^T$의 차원수가 m-r 이다.\n",
    "\n",
    "- Dim(C($A^T$)) + Dim(N(A)) = n\n",
    "\n",
    "A의 row space의 차원수가 r 이고, A의 Null space가 n-r 이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Orthogonal Complement Subspace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$v \\in V, w\\in W$ in $R^n$ 일 때,\n",
    "\n",
    "벡터 v와 w가 $v \\perp w$ orthogonal subspace이면\n",
    "\n",
    "벡터 v와 w의 차원수의 합은 n이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- orthogonal subspace $\\neq$ orthogonal complement subspace"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cosines and Projctions onto lines"
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위 와 같은 그림이 있을 때, cosines 제 2법칙을 이용하면, \n",
    "\n",
    "$$||b-a||^2 = ||a||^2 + ||b||^2 - 2||a|| ||b||cos\\theta$$\n",
    "\n",
    "\n",
    "$$(b-a)^T(b-a) = a^Ta + b^Tb-2||a||||b||cos\\theta$$\n",
    "\n",
    "\n",
    "$$a^Tb = ||a||||b||cos\\theta$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "가 되고, 즉, 벡터 a와 b의 내적은 a의길이, b의길이, 두 벡터가 이루는 각의 cosine 값의 곱이 된다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위 식의 의미는 두 벡터의 a, b의 내적은 서로 다른 벡터에 projection 한다고 할 수 있다."
   ]
  },
  {
   "attachments": {
    "image-2.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image-2.png](attachment:image-2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "projection(사영)을 위와 같은 그림으로 나타 낼수 있는데, 벡터 p와 a는 서로 수직이기 때문에 사영의 길이 $\\hat{x}$ 는 다음과 같이 증명이 가능하다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$(b-\\hat{x}a) \\perp a \\\\ a^T(b-\\hat{x}a) = 0 \\\\ \\hat{x} = {a^Tb\\over a^Ta}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "여기서, $p=\\hat{x}a = {aa^T \\over a^Ta}$ 이고, $p^2 = {aa^Taa^T \\over (a^Ta)(a^Ta)} = {aa^T \\over a^Ta} = p$ 이다.\n",
    "\n",
    "수학적으로, 어떠한 수를 거듭제곱해서 자기자신이 되는 모든 수를 사영한다고 한다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Projections and Least Squares"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "식이 미지수의 개수 보다많은 경우($A_{m \\times n} 에서 m>n인 경우$) $||Ax-b||$^2가 최소가되는 값을 찾아야한다. 이러한 방법을 Least Square method라고 한다."
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위 그림처럼 $Ax \\neq b$이기 때문에 $A$의 Column Space안에 b가 존재하지 않는다. 따라서 벡터 $A\\hat{x}$에서 b에 가장근접한 값, 즉, $b$와 수직이 되는 벡터를 찾아야 한다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "즉, 벡터 $b-A\\hat{x} \\perp a_i$(A의 column vectors)를 만족하는 b인 것이다. 이 식을 모든 n에 대해서 나열하면\n",
    "\n",
    "$$a_1^T(b-A\\hat{x}) = 0$$\n",
    "$$a_2^T(b-A\\hat{x}) = 0$$\n",
    "$$a_3^T(b-A\\hat{x}) = 0$$\n",
    "$$\\vdots$$\n",
    "$$a_n^T(b-A\\hat{x}) = 0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위와 같이 되는데 이를 일반화 하면\n",
    "\n",
    "$$ A^T(b-A\\hat{x}) = 0 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "가 된다. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "즉,"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$A^TA\\hat{x} = A^Tb \\\\\n",
    "\\hat{x} = (A^TA)^{-1}A^Tb \\\\\n",
    "p = A\\hat{x} = A(A^TA)^{-1}A^{T}b$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "가 된다. 이때 $A^TA$가 존재하면 해를 쉽게 구할수 있다. 이때의 값을 best estimate라고 한다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example 1) $A=\\begin{bmatrix} 1&2 \\\\ 1&3 \\\\ 0&0 \\end{bmatrix}$, $b = \\begin{bmatrix} 4 \\\\ 5 \\\\ 6 \\end{bmatrix}$를 $x-y$ plane에 투영하면 다음과 같이 된다. "
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "행렬 A를 구성하는 column vector들이 서로 선형 독립이기 때문에 $A^TA$의 역행력 값을 구할 수 있다. 따라서 P의 값은 아래와 같다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$A^TA = \\begin{bmatrix} 1&1&0 \\\\ 2&3&0 \\end{bmatrix}\\begin{bmatrix} 1&2 \\\\ 1&3 \\\\ 0&0 \\end{bmatrix}=\\begin{bmatrix} 2&5 \\\\ 5&13 \\end{bmatrix}$$\n",
    "\n",
    "$$P = A(A^TA)^{-1}A^T $$\n",
    "$$= \\begin{bmatrix} 1&2 \\\\ 1&3 \\\\ 0&0 \\end{bmatrix}\\begin{bmatrix} 13&-5 \\\\ -5&2 \\end{bmatrix}\\begin{bmatrix} 1&1&0 \\\\ 2&3&0 \\end{bmatrix}$$\n",
    "\n",
    "$$Pb = \\begin{bmatrix} 4 \\\\ 5 \\\\ 0 \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Least Square for line fitting"
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "위와 같은 세 점이 존재할 때, 세 점을 동시에 지나는 직선은 존재 하지 않는다. $y=ax+b$의 형태의 식에 각 점의 값들을 대입하면,\n",
    "\n",
    "$$y_1 = ax_1+b$$\n",
    "$$y_2 = ax_2+b$$\n",
    "$$y_3 = ax_3+b$$\n",
    "$$\\vdots$$\n",
    "$$y_n = ax_n+b$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "가 되고 행렬로 나타내면,\n",
    "\n",
    "$$\\begin{bmatrix} x_1 & 1 \\\\ x_2 & 1 \\\\ x_3 & 1 \\\\ \\vdots \\\\ x_n & 1\\end{bmatrix} \\begin{bmatrix} a \\\\ b \\end{bmatrix} = \\begin{bmatrix} y_1 \\\\ y_2 \\\\ \\vdots \\\\ y_n \\end{bmatrix}$$\n",
    "\n",
    "이와 같이 Ax=b 꼴로 나타낼 수 있다. 따라서, $||Ax-b||^2$를 정확하게 구하기 힘들기 때문에 $min||Ax-b||^2$ 값을 구해야 한다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$Ax-b = \\begin{bmatrix} ax_1+b-y_1 \\\\ ax_2+b-y_2 \\\\ \\vdots \\\\ ax_n+b-y_n \\end{bmatrix} = \\sum_{i=1}^n (ax_i+b-y_i)^2$ 이고 그림으로 나타내면 아래와 같다."
   ]
  },
  {
   "attachments": {
    "image-2.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image-2.png](attachment:image-2.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "즉,$min||Ax-b||^2$은 파란점선이 최소가 되는 직선을 찾는 것이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Grom-Schmidit Orthogonalization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "만약 서로 독립적인 벡터 $a_1,a_2,a_3,...$이 주어져 있으면 Grom-Schmidit Orthogonalization 방법을 이용하여 서로 수직이면서 크기가 1인 basis vector들을 구할 수 있다. Grom-Schmidit Orthogonalization 방법은 다음과 같다."
   ]
  },
  {
   "attachments": {
    "image.png": {
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    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1) 그림과 같이 벡터 a,b가 주어져 있을 때, 벡터 a에 대하여 크기가 1로 만들어 주는 normalization을 하고 $q_1$을 만들어 준다.\n",
    "\n",
    "$$a \\rightarrow {a \\over ||a||} = q_1$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2) 벡터 b를 $q_1$에 project(사영) 한다."
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "project를 한 벡터는 \n",
    "\n",
    "$${q_1^Tb \\over q_1^Tq_1}q_1$$\n",
    "\n",
    "\n",
    "\n",
    "이 되고, q_1은 크기가 1인 벡터이기 때문에 $q_1^Tq_1=1$ 이다. 따라서 $(q_1^Tb)q_1$만 남게 된다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "그러면 $$b-(q_1^Tb)q_1 \\perp q_1$$인것을 확인 할 수 있다."
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3) 벡터 $b-(q_1^Tb)q_1$에 대하여 크기가 1로 만들어 주는 normalization을 한 $q_2$ 벡터를 만들어 준다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$${b-(q_1^Tb)q_1 \\over ||b-(q_1^Tb)q_1||} = q_2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4) $q_2$벡터에 b 벡터를 사영한다."
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "그러면 vector b는 평행사변형법을 이용하여 계산할 수 있다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$b =(q_1^Tb)q_1+(q_2^Tb)q_2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Grom-Schmidit Orthogonalization 방법을 다시 정리하면 다음과 같다.\n",
    "\n",
    "서로 독립인 벡터 $a_1,a_2,...a_n$이 있을 때, 다음순서를 이용하여 푼다.\n",
    "\n",
    "1) $q_1 = {a \\over ||a||}$\n",
    "\n",
    "2) $a_j - \\sum_{j=1}^{i-1}(q_i^Ta_j)q_j $\n",
    "\n",
    "3) ${A_j \\over ||A_j||} = q_j$\n",
    "\n",
    "q의 개수가 차원수만큼 나올 때 까지 2번과 3번을 반복한다.\n",
    "\n",
    "최종적으로 $a_j = \\sum_{i=1}^j(q_i^Ta_j)q_i$를 구하게 된다.\n",
    "\n",
    "코드로 구현하면 다음과 같다."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00, -3.12693935e-16, -3.84592537e-16,\n",
       "        -1.38777878e-17,  0.00000000e+00, -5.12790050e-16],\n",
       "       [-3.12693935e-16,  1.00000000e+00,  1.41308321e-15,\n",
       "         1.11022302e-16,  6.28036983e-16,  6.28036983e-16],\n",
       "       [-3.84592537e-16,  1.41308321e-15,  2.21867130e-30,\n",
       "         1.02786679e-16,  8.87468518e-31,  1.13398755e-30],\n",
       "       [-1.38777878e-17,  1.11022302e-16,  1.02786679e-16,\n",
       "         1.00000000e+00,  7.39557099e-32,  6.85244528e-17],\n",
       "       [ 0.00000000e+00,  6.28036983e-16,  8.87468518e-31,\n",
       "         7.39557099e-32,  3.94430453e-31,  3.94430453e-31],\n",
       "       [-5.12790050e-16,  6.28036983e-16,  1.13398755e-30,\n",
       "         6.85244528e-17,  3.94430453e-31,  6.90253292e-31]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "def gram_schmidt(X):\n",
    "    O = np.zeros(X.shape)\n",
    "    for i in range(X.shape[1]):\n",
    "        # orthogonalization\n",
    "        vector = X[:, i]\n",
    "        space = O[:, :i]\n",
    "        projection = vector @ space\n",
    "        vector = vector - np.sum(projection * space, axis=1)\n",
    "        # normalization\n",
    "        norm = np.sqrt(vector @ vector)\n",
    "        vector /= abs(norm) < 1e-8 and 1 or norm\n",
    "        \n",
    "        O[:, i] = vector\n",
    "    return O\n",
    "\n",
    "vectors = np.array([\n",
    "    [1, 1, 2, 0, 1, 1],\n",
    "    [0, 0, 0, 1, 2, 1],\n",
    "    [1, 2, 3, 1, 3, 2],\n",
    "    [1, 0, 1, 0, 1, 1]\n",
    "    ], dtype=float)\n",
    "orthonormal = gram_schmidt(vectors)\n",
    "\n",
    "orthonormal.T @ orthonormal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Orthogonal Basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "서로 직교 하는 벡터들을 가지고 벡터공간을 span하는것을 Orthogonal Basis라고 한다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$q_1,q_2,...,q_n$이 orthonormal(서로 수직이면서 크기가 1이다.) 한다고 하자."
   ]
  },
  {
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    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![image.png](attachment:image.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "열벡터 $q$들로 만든 행렬을 Q라고 했을때, $Q^TQ = I \\Rightarrow Q^T=Q^{-1}$를 만족한다. 즉, $Q^T$ 는 Q의 left-inverse이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Function Space(Hilbert Space)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Function Space는 벡터 공간을 여러가지 성질을 함수공간에 확장 하는것이다. n차원의 Vector Space를 무한차원의 Hilbert Space로 확장하는 것이다. 즉, 무리수같은 real number공간들도 생각해서 vector space를 생각해본다는 것이다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "다음과 같이 구간이 정해져있는 함수들의 공간이 존재할 때,\n",
    "\n",
    "$$x(t),y(t) \\in \\mathcal{H}\\ a \\le t \\le b$$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "다음과 같은 성질을 만족한다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- $x(t)+y \\in \\mathcal{H}$ (덧셈에 닫혀있다.)\n",
    "\n",
    "- $\\alpha x(t) \\in \\mathcal{H}$ (곱셈에 닫혀있다.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 1) space 표현\n",
    "\n",
    "vector space를 다음과 같이 표현할 수 있으면, Vector : $V = (v_1,v_2,...,v_n)^T$ \n",
    "\n",
    "Hilbert Space는 다음과 같이 표현한다. $lim_{\\Delta t \\to 0} x(t) = (x(a),x(a+\\Delta t),x(a+2\\Delta t),...,x(b))^T \\to x(t) \\in R^\\infty$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 2) 내적 표현"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "vector inner product\n",
    "\n",
    "$$x^Ty= \\sum_{i=1}^n x_iy_i$$\n",
    "\n",
    "function inner product \n",
    "\n",
    "$$(x(t),y(t)) =lim_{\\Delta t \\to 0}\\sum_{k=0}^\\infty x(a+k\\Delta t)y(a+k\\Delta t) \\\\\n",
    "=\\int_a^b x(t)y(t)dt \\\\\n",
    "=||x(t)||^2 = \\int_a^b x^2(t)dt$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3) orthogonality"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "vetor\n",
    "\n",
    "$$X^Ty =0$$\n",
    "\n",
    "function\n",
    "\n",
    "$$(x(t),y(t))=\\int_a^b x(t)y(t)dt=0$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 4)  Basis\n",
    "\n",
    "vector\n",
    "\n",
    "$$X=\\sum_{i=1}^na_iv_i$$\n",
    "\n",
    "function\n",
    "\n",
    "$$x(t)=\\sum_{i=1}^\\infty a_ib_i(t)$$\n",
    "\n",
    "vector 공간에서는 basis들의 합성곱을 선형결합이라고 하지만 function  공간에서는 Series 라고한다."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5) $a_i$ 구하는법(Grom-Schmidit Orthogonalization)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "vector\n",
    "\n",
    "$$\\sum_{i=1}^n(g_i^Ta)g_i$$\n",
    "\n",
    "functioon\n",
    "\n",
    "$$\\sum_{i=1}^\\infty (g_i(t),x(t))g_i(t)$$\n",
    "\n",
    "벡터 공간처럼 함수공간도 $a_i$의 series coefficients 값들은 unique하다."
   ]
  }
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