{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", " 線形代数(Linear Algebra IV)写像 \n", "
\n", "
\n", " cc by Shigeto R. Nishitani, 2018-03-19 \n", "
\n", "\n", "* file: /Users/bob/python/doing_math_with_python/linear_algebra/LA-V_map.ipynb" ] }, { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

1  写像
1.1  線形写像
1.2  像と核
1.3  不変部分空間
2  紙
3  python code
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 写像\n", "\n", "## 線形写像\n", "* $n$次元数ベクトル空間$\\boldsymbol{R}^n$から$m$次元数ベクトル空間$\\boldsymbol{R}^m$への写像$f$が\n", "\n", " 1. $\\boldsymbol{R}^n$の任意のベクトル$ \\boldsymbol{a}, \\boldsymbol{b} $に対し，$f( \\boldsymbol{a} + \\boldsymbol{b} ) = f( \\boldsymbol{a} ) + f( \\boldsymbol{b} )$\n", " 1. $\\boldsymbol{R}^n$の任意のベクトル$ \\boldsymbol{a}$と任意の実数$\\lambda$に対し，$f(\\lambda \\boldsymbol{a} )=\\lambda f( \\boldsymbol{a} )$\n", "\n", "が成り立つとき，$f$を$\\boldsymbol{R}^n$から$\\boldsymbol{R}^m$への**線形写像**という．\n", "\n", "* 零ベクトルは零ベクトルへ写像される($f(\\boldsymbol{0})=\\boldsymbol{0}$)\n", "* $f(-\\boldsymbol{a})=-\\boldsymbol{a}$\n", "* 零写像や恒等写像も線形写像\n", "\n", "* **線形写像の表現行列**\n", "$\\boldsymbol{e}_1, \\boldsymbol{e}_2, \\ldots, \\boldsymbol{e}_n$を$R^n$の標準的な基底とする．\n", "$$\n", "f(\\boldsymbol{e}_j) = \\boldsymbol{a}_j = (\\boldsymbol{a}_{1j}, \\boldsymbol{a}_{2j},\\ldots,\\boldsymbol{a}_{mj})\\, (j=1,2,\\ldots,n)\n", "$$\n", "のときこれらを列ベクトルみなして,$m \\times n$行列\n", "$$\n", "A = [f(\\boldsymbol{e}_1)\\, f(\\boldsymbol{e}_2)\\, \\cdots\\, f(\\boldsymbol{e}_n)]=\n", "[a_1\\, a_2\\, \\cdots \\, a_n]=\n", "\\left(\n", "\\begin{array}{cccc}\n", "a_{11} & a_{12} & \\cdots & a_{1n} \\\\\n", "a_{21} & a_{22} & \\cdots & a_{2n} \\\\\n", "\\vdots & \\vdots & \\ddots & \\vdots \\\\\n", "a_{m1} & a_{m2} & \\cdots & a_{mn} \\\\\n", "\\end{array}\n", "\\right)\n", "$$\n", "を$f$の**(標準的な基底に関する)表現行列**という．\n", "\n", "* $\\boldsymbol{y}=f(\\boldsymbol{x})$の時，\n", "$$\n", "\\left(\n", "\\begin{array}{c}\n", "y_1 \\\\\n", "y_2 \\\\\n", "\\vdots \\\\\n", "y_m\n", "\\end{array}\n", "\\right)\n", "= A\n", "\\left(\n", "\\begin{array}{c}\n", "x_1 \\\\\n", "x_2 \\\\\n", "\\vdots \\\\\n", "x_n\n", "\\end{array}\n", "\\right)\n", "$$\n", "\n", "* $\\boldsymbol{R}^n$から$\\boldsymbol{R}^m$への写像\n", "$$\n", "(x_1,x_2,\\ldots,x_n) \\longmapsto (y_1,y_2,\\ldots,y_m)\n", "$$\n", "\n", "* $\\boldsymbol{x}\\in \\boldsymbol{R}^n$を列ベクトルとみなせば，\n", "$$\n", "f( \\boldsymbol{x} ) = A \\boldsymbol{x} \n", "$$\n", "が成り立つ．\n", "\n", "* 線形写像の和・実数倍・合成\n", "$$\n", "(f+g)(\\boldsymbol{x}) = f(\\boldsymbol{x}) + g(\\boldsymbol{x}) \\\\\n", "(\\lambda f) \\boldsymbol{x} = \\lambda f(\\boldsymbol{x}) \\\\\n", "(g \\circ f)(\\boldsymbol{x}) = g(f(\\boldsymbol{x}))\n", "$$\n", "$g \\circ f$を合成写像または積という．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 像と核\n", "$f$を$\\boldsymbol{R}^n$から$\\boldsymbol{R}^m$への線形写像とする．\n", "* **像(Im)**\n", "${\\rm Im\\,}f = \\{f( \\boldsymbol{x} ); \\boldsymbol{x} \\in \\boldsymbol{R}^n\\}$は$\\boldsymbol{R}^m$の部分空間でこれを**像(空間)**という．\n", "$f$の表現行列を$A=[\\boldsymbol{a}_1\\, \\boldsymbol{a}_2\\, \\cdots\\, \\boldsymbol{a}_n]$とする時，\n", " * ${\\rm Im\\,}f =L\\{\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\ldots,\\boldsymbol{a}_n\\}$($\\boldsymbol{a}_1,\\boldsymbol{a}_2,\\ldots,\\boldsymbol{a}_n$で生成される部分空間)\n", " * ${\\rm dim\\,}({\\rm Im\\,}f) = {\\rm rank\\,} A = {\\rm rank\\,}[[\\boldsymbol{a}_1\\,\\boldsymbol{a}_2\\,\\cdots\\,\\boldsymbol{a}_n]$\n", " * $\\boldsymbol{y}\\in {\\rm Im}f \\iff\n", "{\\rm rank\\,}[\\boldsymbol{a}_1\\,\\boldsymbol{a}_2\\,\\cdots\\,\\boldsymbol{a}_n]=\n", "{\\rm rank\\,}[\\boldsymbol{a}_1\\,\\boldsymbol{a}_2\\,\\cdots\\,\\boldsymbol{a}_n \\boldsymbol{y}]\n", "$\n", "\n", "* **全射(onto mapping)**\n", "${\\rm Im\\,}f = \\boldsymbol{R}^m$が全射であるとは，\n", " * $すべての\\boldsymbol{y}\\in \\boldsymbol{R}^m \\Longrightarrow \n", "f( \\boldsymbol{x} ) = \\boldsymbol{y} なる \\boldsymbol{x} \\in R^n が存在する$\n", " * $fが全射 \\iff {\\rm rank\\,}A = m$\n", "* **核(Ker)**\n", "${\\rm Ker\\,}f = \\{\\boldsymbol{x}\\in \\boldsymbol{R}^n; f( \\boldsymbol{x} )= \\boldsymbol{0} \\}$は$\\boldsymbol{R}^n$の部分空間であってこれを**核(空間)**という.\n", " * ${\\rm Ker\\,}f=\\{\\boldsymbol{x};A \\boldsymbol{x} = \\boldsymbol{0} \\}$:同次連立一次方程式$A \\boldsymbol{x} = \\boldsymbol{0}$の解空間\n", " * ${\\rm dim\\,}({\\rm Ker\\,} f)=n -{\\rm rank\\,} A$\n", "* **単射(one to one mapping)**\n", "${\\rm Ker\\,}f=\\{\\boldsymbol{0}\\}$のとき$f$を**単射**であるという．このとき，\n", " * $f(\\boldsymbol{x}_1)=f(\\boldsymbol{x}_2) \\Longrightarrow\n", "\\boldsymbol{x}_1 = \\boldsymbol{x}_2$\n", " * $fが単射 \\iff {\\rm rank\\,} A = n$\n", "* **次元定理**\n", "$ {\\rm dim\\,}({\\rm Im\\,}f)+{\\rm dim\\,}({\\rm Ker\\,}f) = n$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 不変部分空間\n", "\n", "$f$を$\\boldsymbol{R}^n$上の線形変換とする．$\\boldsymbol{R}^n$の部分空間$V$が\n", "$$\n", "f(V) \\subset V\n", "$$\n", "をみたすとき，$V$を$f$に関する**不変部分空間**という．\n", "\n", "* 例\n", " * 核${\\rm Ker\\,}f$: $\\boldsymbol{x} \\in {\\rm Ker\\,}f \\Rightarrow f(\\boldsymbol{x})=\\boldsymbol{0} \\in {\\rm Ker\\,} fよってf({\\rm Ker\\,} f)\\subset {\\rm Ker\\,}f$\n", " * 像${\\rm Im\\,}f$: $\\boldsymbol{x} \\in {\\rm Im\\,}f \\Rightarrow f(\\boldsymbol{x}) \\in {\\rm Im\\,} fよってf({\\rm Im\\,} f)\\subset {\\rm Im\\,}f$\n", " * 同次連立一次方程式の解空間: $\\{\\boldsymbol{x}\\in \\boldsymbol{R}^m; A\\boldsymbol{x}=\\boldsymbol{0}\\}(= {\\rm Ker\\,} f)$\n", " * 固有値$\\lambda$に対する固有空間: $\\{\\boldsymbol{x}\\in\\boldsymbol{R}^n;(A-\\lambda E)\\boldsymbol{x}=\\boldsymbol{0}\\}$\n", " * 固有値$\\lambda$に対する一般固有空間($k$は指標): $\\{\\boldsymbol{x}\\in\\boldsymbol{R}^n;(A-\\lambda E)^k\\boldsymbol{x}=\\boldsymbol{0}\\}$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 紙\n", "\n", "![LA-V](./figs/LinearAlgebra-V.png)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# python code" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "array([ 4., 0.])\n", "array([[ 0.447, -0.447],\n", " [ 0.894, 0.894]])\n", "array([ 0.447, 0.894])\n", "array([-0.447, 0.894])\n" ] } ], "source": [ "import numpy as np\n", "from pprint import pprint\n", "import scipy.linalg as linalg\n", "\n", "np.set_printoptions(precision=3, suppress=True)\n", "\n", "a = np.array([[2,1], [4,2]])\n", "# a = np.array([[2,1], [1,2]])\n", "\n", "l,P = np.linalg.eig(a)\n", "pprint(l)\n", "pprint(P)\n", "\n", "v0 = P[:,0]\n", "v1 = P[:,1]\n", "pprint(v0)\n", "pprint(v1)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "\n", "div = 32\n", "for i in range(0,div):\n", " theta = 2*np.pi/div*i\n", " x = np.sin(theta)\n", " y = np.cos(theta)\n", " # print('%10.5f-%10.5f' % (x,y))\n", " plt.plot(x,y,'o',color='r')\n", " p0 = np.array([x,y])\n", " p1 = np.dot(a,p0)\n", " plt.plot(p1[0],p1[1],'o',color='b')\n", " plt.plot([x, p1[0]], [y,p1[1]], color='k', linestyle='-', linewidth=1)\n", "\n", "x_m = 6\n", "y_m = 6\n", "plt.hlines(0, -x_m, x_m, color='k', linestyle='-', linewidth=1)\n", "plt.vlines(0, -y_m, y_m, color='k', linestyle='-', linewidth=1)\n", "\n", "t=x_m\n", "plt.plot([-t*v0[0],t*v0[0]], [-t*v0[1],t*v0[1]], color='g', linestyle='-', linewidth=2)\n", "t=y_m\n", "plt.plot([-t*v1[0],t*v1[0]], [-t*v1[1],t*v1[1]], color='g', linestyle='-', linewidth=2)\n", "\n", "\n", "plt.axes().set_aspect('equal', 'datalim')\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, 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