{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 1.2 데이터의 형식과 기본 연산 (part 2)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 벡터의 표현과 내적" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 벡터의 표현" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "### `성분의 배열`으로서의 벡터 vs `크기와 방향을 가지는` 물리적 벡터" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "앞에서 **벡터**를 여러 개의 성분이 가로 또는 세로로 배열된 것으로 정의하고, 1행 또는 1열짜리 행렬으로 보았다. \n", "이러한 정의는 주로 컴퓨터 과학 분야에서 사용된다. \n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "똑같은 개념의 벡터를 수학이나 물리학에서는 \n", "보통 성분을 먼저 이야기하지 않고, 아래와 같이 크기와 방향을 가지는 양이라고 정의한다. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "이후에 **위치벡터**라는 개념을 소개하여, 유클리드 공간 $\\mathbb{R}^n$의 벡터를 성분을 이용하여 표현할 수 있도록 한다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### `크기와 방향을 가지는` 물리적 벡터" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 주어진 공간의 점 $P$에서 시작하여 점 $Q$에서 끝나는 벡터를 기호 $\\overrightarrow{PQ}$로 나타낸다. \n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 이때 점 $P$를 벡터의 **시점(initial point)**, 점 $Q$를 **종점(terminal point)** 이라고 한다. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 이 **벡터의 크기** 는 기호 $\\Vert\\overrightarrow{PQ}\\Vert$로 나타내며, 주어진 공간에서의 점 $P$와 점 $Q$ 사이의 거리를 뜻한다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 크기가 $1$인 벡터를 **단위벡터(unit vector)**, 크기가 $0$인 벡터를 **영벡터**라 한다. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### 같은 벡터의 정의" ] }, { "attachments": { "vectors-equal.JPG": { "image/jpeg": 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} }, "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "
\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 벡터가 놓인 위치에 상관없이, 크기와 방향이 같으면 **같은 벡터** 라고 한다. \n", "\n", "\n", "* 한 벡터를 평행이동한 것은 모두 같은 벡터이다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 벡터를 한 문자로 나타낼 때는 주로 굵은 소문자 알파벳을 사용한다. \n", "예를 들어, ${\\bf v}=\\overrightarrow{AB}=\\overrightarrow{CD}$ 와 같이 표현한다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### 벡터의 스칼라곱" ] }, { "attachments": { "vector-scalar-multiples.JPG": { "image/jpeg": 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} }, "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "
\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 벡터 $\\mathbf{v}$에 스칼라(실수) $k$를 곱한 벡터 $k{\\bf v}$의 \n", " 크기는 $\\|k{\\bf v}\\|=|k|\\,\\|{\\bf v}\\|$이고, \n", " 양의 실수를 곱하면 같은 방향, 음의 실수를 곱하면 반대 방향을 나타내는 것으로 약속한다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 크기가 같고 방향이 반대인 벡터를 $-$ 부호를 붙여 나타낸다. 즉, $-\\overrightarrow{PQ}=\\overrightarrow{QP}\\,$이다. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* $\\mathbf{v}$가 영벡터가 아닐 때, $\\dfrac{1}{\\|\\mathbf{v}\\|}\\,\\mathbf{v}=\\dfrac{\\mathbf{v}}{\\|\\mathbf{v}\\|}$는 $\\mathbf{v}$ 방향의 단위벡터이다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 두 벡터가 평행일 필요충분조건은 한 벡터가 다른 벡터의 스칼라 곱인 것이다. \n", " 즉, ${\\bf v}\\parallel {\\bf w} \\ \\ \\iff \\ \\ {\\bf v}=k{\\bf w} \\ \\ (k\\mbox{는 실수})$." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### 벡터의 합" ] }, { "attachments": { "vectors-sum.JPG": { "image/jpeg": 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} }, "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "
\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 두 벡터 ${\\bf v}$와 ${\\bf w}$에 대하여 **벡터의 합** ${\\bf v}+{\\bf w}$는 \n", "${\\bf v}$의 종점에 ${\\bf w}$의 시점을 일치시켰을 때, \n", "${\\bf v}$의 시점에서 ${\\bf w}$의 종점으로 가는 벡터를 뜻한다. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 즉, ${\\bf v}=\\overrightarrow{AB}$, ${\\bf w}=\\overrightarrow{BC}$ 이면 ${\\bf v}+{\\bf w}=\\overrightarrow{AB}+\\overrightarrow{BC}=\\overrightarrow{AC}$ 이다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "
\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 평행이동한 벡터는 모두 같음을 이용하면 ${\\bf v}+{\\bf w}={\\bf w}+{\\bf v}$가 항상 성립 \n", "\n", "\n", "* ${\\bf v}$와 ${\\bf w}$의 시점을 일치시켜서 나타내면 ${\\bf v}+{\\bf w}$는 같은 시점에서 출발하는 평행사변형의 대각선 벡터 \n", "\n", "\n", "* ${\\bf v}-{\\bf w}={\\bf v}+(-{\\bf w})$로 생각하면 ${\\bf v}-{\\bf w}$는 삼각형의 한 변으로서 ${\\bf w}$의 종점에서 ${\\bf v}$의 종점으로 가는 방향의 벡터" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### `모든 벡터의 시작점을 일치시켜 나타내는` 위치 벡터" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "이제 3차원 유클리드 공간 $\\mathbb{R}^3$에서 직교좌표계를 이용하여 벡터를 표현하는 방법을 생각해보자. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "한 벡터 ${\\bf v}$를 공간의 어디에 놓아도 모두 같은 벡터이므로, \n", "모든 벡터의 시점을 정해진 한 점(보통은 원점 $O$)에 두기로 약속하면 \n", "벡터 ${\\bf v}$를 나타낼 때, 그 종점 $P$를 찾아 ${\\bf v}=\\overrightarrow{OP}$ 로 나타내면 된다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "이때 $P$의 좌표가 $(a,b,c)$이면 벡터 $[a,b,c]$를 ${\\bf v}$의 **위치벡터(position vector)**라고 부르고 ${\\bf v}=[a,b,c]$로 나타낸다.\n", "\n", "(표기의 편리를 위하여, 점의 좌표처럼 위치벡터를 ${\\bf v}=(a,b,c)$로 나타내기도 한다.)\n", "\n", "(뒤에서 배울 여러 가지 필요에 따라 $\\mathbb{R}^n$을 행백터들의 집합으로 보거나 또는 열벡터들의 집합으로 보기도 한다.)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "이와 같이, 일반적으로 $\\mathbb{R}^n$의 모든 벡터를 원점 $O$를 기준으로 하는 위치벡터로 나타냄으로써 \n", "수학이나 물리학에서 크기와 방향을 가지는 양으로서의 벡터의 연산을 \n", "다음과 같이 성분이 주어진 벡터의 연산으로 시행할 수 있다. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "**일반적인 $n$차원 유클리드 공간에서의 벡터합과 스칼라곱**\n", "\n", "$\\mathbb{R}^n$의 벡터는 $n$개의 수로 이루어진 순서쌍 $(x_1, x_2, \\cdots, x_n)$으로 표현되고, 이러한 벡터의 합과 스칼라 곱은 \n", "\n", " $$(x_1, x_2, \\cdots, x_n) + (y_1, y_2, \\cdots, y_n) = (x_1 + y_1, x_2 + y_2, \\cdots, x_n + y_n)$$\n", "\n", " $$k(x_1, x_2, \\cdots, x_n) = (kx_1, kx_2, \\cdots, ,kx_n)$$\n", "\n", "으로 정의된다. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 벡터의 내적(inner product)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "$\\mathbb{R}^n$에서 두 벡터 ${\\bf v} = (x_1, x_2, \\cdots, x_n)$와 ${\\bf w}=(y_1, y_2, \\cdots, y_n)$의 **내적**은 \n", "\n", "$$\n", "{\\bf v} \\cdot {\\bf w} = \\sum_{k=1}^n x_k y_k\n", "$$\n", "\n", "으로 정의된다. \n", "\n", "* \\\\(n=2\\\\)인 경우 \\\\[(a, b) \\cdot (x,y) = ax + by\\\\]\n", "\n", "* \\\\(n=3\\\\)인 경우 \\\\[(a,b,c)\\cdot (x,y,z) = ax + by + cz\\\\]" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "**예제 1.1.8**   두 벡터 ${\\bf v}=[1, -1, 0]$, ${\\bf w}=[2, 0, 2]$ 에 대하여 다음을 구하시오. \n", "\n", "(1) ${\\bf v} \\cdot {\\bf w}$         \n", "(2) 벡터 ${\\bf v}$방향의 단위벡터         \n", "(3) 벡터 ${\\bf w}$방향의 단위벡터         " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "**[풀이]**" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "(1) ${\\bf v} \\cdot {\\bf w}=2+0+0=2$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "(2) $\\|\\mathbf{v}\\|=\\sqrt{1^2+(-1)^2+0^2}=\\sqrt{2}$ 이므로   $\\dfrac{1}{\\|\\mathbf{v}\\|}\\,\\mathbf{v}=\\dfrac{1}{\\sqrt{2}}\\,[1, -1, 0]$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "(3) $\\|\\mathbf{w}\\|=\\sqrt{2^2+0^2+2^2}=\\sqrt{8}$ 이므로   $\\dfrac{1}{\\|\\mathbf{w}\\|}\\,\\mathbf{w}=\\dfrac{1}{2\\sqrt{2}}\\,[2, 0, 2]=\\dfrac{1}{\\sqrt{2}}\\,[1, 0, 1]$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "                     \n", "                                               \n", "**[풀이 끝]** \n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "**파이썬 예제 1.1.9**   \n", "python에서 내적은 numpy.inner(a,b), 또는 np.dot(a,b)를 이용해 계산할 수 있다. \n", "\n", "참고로 비슷한 다른 표현과의 차이를 살펴보자." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "**[풀이]**" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "text/plain": [ "28" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "a=np.array([1,2,3])\n", "b=np.array([2,4,6])\n", "np.inner(a,b)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "scrolled": true, "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "text/plain": [ "28" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.dot(a,b)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "text/plain": [ "array([2, 4, 6])" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "2*a" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "text/plain": [ "array([ 2, 8, 18])" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a*b" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "text/plain": [ "[1, 2, 3, 1, 2, 3]" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "2*[1,2,3]" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "                     \n", "                                               \n", "**[풀이 끝]** \n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 벡터의 내적이 갖고 있는 성질" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "(1)   ${\\bf v} \\cdot {\\bf w} = {\\bf w} \\cdot {\\bf v}$\n", "\n", "(2)   ${\\bf v}\\cdot({\\bf w}_1 + {\\bf w}_2) = {\\bf v}\\cdot {\\bf w}_1 + {\\bf v}\\cdot {\\bf w}_2 $\n", "\n", "(3)   $k({\\bf v}\\cdot {\\bf w}) = (k{\\bf v})\\cdot {\\bf w} = {\\bf v}\\cdot (k{\\bf w})$       ($k$는 실수)\n", "\n", "(4)   ${\\bf v}\\cdot {\\bf v}= \\Vert {\\bf v}\\Vert^2$ \n", "\n", "(5)   $\\mathbf{0}\\cdot {\\bf v} = 0$       (여기에서 $\\mathbf{0}$은 영벡터, $0$은 실수 영을 나타낸다.)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 두 벡터의 사잇각과 내적" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "두 벡터 ${\\bf v}$, ${\\bf w}$에 대하여 \n", "그 사잇각을 $\\theta$  ($0 \\le \\theta \\le \\pi$) 라고 하면\n", "$$\n", " {\\bf v}\\cdot {\\bf w} = \\Vert {\\bf v} \\Vert \\Vert\\, {\\bf w} \\Vert \\cos \\theta\n", "$$\n", "가 성립한다. (증명은 생략) \n", "\n", "따라서 $\\cos \\theta=\\dfrac{{\\bf v}\\cdot {\\bf w}}{\\Vert {\\bf v} \\Vert \\Vert\\, {\\bf w} \\Vert}$ 를 구하여 두 벡터 사이의 각 $\\theta$를 찾을 수 있다.\n", "\n", "그리고 ${\\bf v}\\cdot {\\bf w} = 0$ 이면 ${\\bf v}\\perp {\\bf w}$  (서로 수직)임을 알 수 있다.   (영벡터는 모든 벡터와 수직이다.)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "**예제 1.1.10**   세 벡터 ${\\bf u}=[3, 3, 1]$, ${\\bf v}=[1, -1, 0]$, ${\\bf w}=[2, 0, 2]$ 에 대하여 다음 물음에 답하시오. \n", "\n", "(1) ${\\bf v}$, ${\\bf w}$ 사이의 각 $\\theta$를 구하시오. \n", "\n", "(2) ${\\bf u}$, ${\\bf v}$ 는 서로 수직인가?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "**[풀이]**" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "(1) 위의 예제 1.1.8의 결과로부터 $\\cos \\theta=\\dfrac{{\\bf v}\\cdot {\\bf w}}{\\Vert {\\bf v} \\Vert \\Vert\\, {\\bf w} \\Vert}\n", "= \\dfrac2{\\sqrt{2}\\,\\sqrt{8}}=\\dfrac12$ 이므로 $\\theta=\\dfrac{\\pi}3$ 이다.   ($0 \\le \\theta \\le \\pi$)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "(2) ${\\bf u}\\cdot {\\bf v}=3-3+0=0$ 이므로 ${\\bf u}$, ${\\bf v}$ 는 서로 수직이다." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "                     \n", "                                               \n", "**[풀이 끝]** \n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "
\n", "\n", "## 1.1 (part 2) 연습문제 과제" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 먼저 이 강의의 jupyter notebook 파일에서 예제 1.1.8, 파이썬 예제 1.1.9, 예제 1.1.10 부분만 뽑아서 새로운 파일로 저장하세요." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "fragment" } }, "source": [ "* 과제는 이 문제들 각각에 대하여 일부분(숫자, 문자, 함수 등)을 바꾸어 새로운 문제를 만들고 풀이를 써서 jupyter notebook 파일로 제출하는 것입니다. " ] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.8" } }, "nbformat": 4, "nbformat_minor": 4 }