{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", " 線形代数(Linear Algebra III) ベクトル空間 \n", "
\n", "
\n", " cc by Shigeto R. Nishitani, 2018-03-16 \n", "
\n", "\n", "* file: /Users/bob/python/doing_math_with_python/linear_algebra/LA-III_vector_space.ipynb" ] }, { "cell_type": "markdown", "metadata": { "toc": "true" }, "source": [ "# Table of Contents\n", "

1  定義と解説
1.1  実数上の$n$次(元)数ベクトル空間
1.2  線形結合
1.3  基底，次元，要素
1.4  部分空間
1.5  計量ベクトル空間
1.6  直交基底
1.7  直交補空間(orthogonal complement, perp)
2  紙
3  python
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 定義と解説\n", "\n", "## 実数上の$n$次(元)数ベクトル空間\n", "実数の$n$個の組の全体\n", "$$\n", "\\boldsymbol{R}^n = \\{ (a_1,a_2,\\ldots,a_n); a_1, a_2, \\ldots, a_n \\in \\boldsymbol{R} \\}\n", "$$\n", "\n", "* 定義\n", "$\\boldsymbol{a}=(a_1,a_2,\\ldots,a_n), \\boldsymbol{b}=(b_1,b_2,\\ldots,b_n) \\in \\boldsymbol{R}^n$に対して，\n", " * **相等** $\\boldsymbol{a}=\\boldsymbol{b} \\iff a_1=b_1, a_2=b_2, \\ldots , a_n = b_n$\n", " * **和** $\\boldsymbol{a}+\\boldsymbol{b} =( a_1+b_1, a_2+b_2, \\ldots , a_n + b_n)$\n", " * **スカラー倍** $\\lambda\\boldsymbol{a}= (\\lambda a_1, \\lambda a_2, \\ldots , \\lambda a_n)$\n", "\n", "* 数ベクトル空間の要素を**数ベクトル**あるいは単に**ベクトル**という．\n", "* 数ベクトル$\\boldsymbol{a}$は$n$次元行ベクトルと見なせるし，\n", "$$\n", "\\boldsymbol{a} = \\left(\\begin{array}{c}\n", "a_1 \\\\\n", "a_2 \\\\\n", "\\vdots \\\\\n", "a_n\n", "\\end{array}\\right)\n", "$$\n", "の$n$次元列ベクトルとも見なせる．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 線形結合\n", "$m$個の$n$次元数ベクトル$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m$\n", "に対して，\n", "$$\n", "x_1 \\boldsymbol{a}_1 + x_2 \\boldsymbol{a}_2 + \\cdots + x_m \\boldsymbol{a}_m,\\,\n", "x_1, x_2, \\ldots, x_m \\in \\boldsymbol{R}\n", "$$\n", "を$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m$の**線形結合(linear combination,一次結合，一次和)**という．\n", "\n", "* **一次独立, 一次従属**\n", "$x_1 \\boldsymbol{a}_1 + x_2 \\boldsymbol{a}_2 + \\cdots + x_m \\boldsymbol{a}_m = \\boldsymbol{0} $となるのが，\n", "$$\n", "\\begin{array}{rcl}\n", "x_1 = x_2 = \\cdots = x_m =0 以外に起こり得ない &\\iff &\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m が一次独立 \\\\\n", "x_1 = x_2 = \\cdots = x_m =0 以外にもあり得る &\\iff &\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m が一次従属\n", "\\end{array}\n", "$$\n", "\n", "* 一次独立性と階数\n", "数ベクトルを列ベクトルで表して$A=[\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m]$を$n \\times m $行列とすると\n", "$$\n", "\\begin{array}{rcl}\n", "\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m が一次独立 &\\iff & {\\rm rank}A = m\\\\\n", "\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m が一次従属 &\\iff & {\\rm rank}A < m\n", "\\end{array}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 基底，次元，要素\n", "* **基底** \n", "$n$次元数ベクトル空間$\\boldsymbol{R}^n$の$n$個の一次独立なベクトル\n", "$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_n$を\n", "$\\boldsymbol{R}^n$の**基底**という\n", "\n", "* **標準的な基底** \n", "$\\boldsymbol{e}_1=(1,0,\\ldots,0),\n", " \\boldsymbol{e}_2=(0,1,\\ldots,0), \\ldots,\n", " \\boldsymbol{e}_n=(0,0,\\ldots,1)$は$\\boldsymbol{R}^n$の基底で，\n", "**$\\boldsymbol{R}^n$の標準的な基底**という．\n", "\n", "* **次元** 基底を構成する個数を**次元**といい，dimで表す．${\\rm dim}\\boldsymbol{R}^n=n$である．\n", "* **成分** \n", " * $\\mathfrak{B}=\\{\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_n\\}$\n", "を$\\boldsymbol{R}^n$の基底とする．$\\boldsymbol{R}^n$のベクトル$\\boldsymbol{a}=\n", "(a_1, a_2, \\ldots, a_n)$は\n", "$$\n", "\\boldsymbol{a} = x_1\\boldsymbol{a}_1+ x_2 \\boldsymbol{a}_2+ \\ldots + x_n\\boldsymbol{a}_n\n", "$$\n", "と一意的に表される．\n", " * この実数の組$(x_1,x_2,\\ldots,x_n)$を$\\boldsymbol{a}$の基底\n", "$\\mathfrak{B}=\\{\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_n\\}$に関する**成分**といい，\n", "$$\n", "\\boldsymbol{a} = (x_1, x_2,\\ldots,x_n)_{\\mathfrak{B}}\n", "$$\n", "とかく．\n", " * この$(a_1, a_2, \\ldots, a_n)$は$\\boldsymbol{a}$の標準的な基底$\\boldsymbol{e}_1, \\boldsymbol{e}_2, \\ldots,\\boldsymbol{e}_n$に関する成分である．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 部分空間\n", "* 数ベクトル空間$\\boldsymbol{R}^n$の部分集合$V(\\neq 0)$が和とスカラー倍に閉じているとき，$\\boldsymbol{R}^n$の**部分空間**という．\n", "\n", "* $\\{\\boldsymbol{0}\\} \\subseteq V \\subseteq \\boldsymbol{R}^n$\n", "* **有限生成な部分空間**\n", "$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m \\in \\boldsymbol{R}^n $に対して$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m$の線形結合全体\n", "$$\n", "L\\{\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m\\}\n", "=\\{x_1\\boldsymbol{a}_1+x_2\\boldsymbol{a}_2+ \\ldots,+x_m\\boldsymbol{a}_m; x_1, x_2, \\ldots, x_m \\in \\boldsymbol{R}\\}\n", "$$\n", "は$\\boldsymbol{R}^n$の部分空間で，これを**$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m \\in \\boldsymbol{R}^n $で生成される(張られる)部分空間**といい，$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m \\in \\boldsymbol{R}^n $を$L\\{\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m \\in \\boldsymbol{R}^n \\}$の**生成系**という．\n", "\n", "* **rank**\n", "$$\n", "\\boldsymbol{b}\\in L\\{\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m\\} \\iff\n", "{\\rm rank}[\\boldsymbol{a}_1\\, \\boldsymbol{a}_2\\, \\ldots\\,\\boldsymbol{a}_m] =\n", "{\\rm rank}[\\boldsymbol{a}_1\\, \\boldsymbol{a}_2\\, \\ldots\\,\\boldsymbol{a}_m\\,\\boldsymbol{b}]\n", "$$ \n", "すなわち，$\\boldsymbol{b}$が生成系に含まれるならば，$\\boldsymbol{b}$を加えて作る拡大係数行列のrankが一致する．(一次従属だから)\n", "\n", "* **基底**\n", "\n", "$$\n", "Vのベクトル\n", "\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m がVの基底\n", "\\iff \n", "\\left\\{\n", "\\begin{array}{l}\n", "(1)\\, \\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m　は一次独立 \\\\\n", "(2)\\, \\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots,\\boldsymbol{a}_m　はVの生成系\n", "\\end{array}\n", "\\right.\n", "$$\n", "\n", "* **同次連立一次方程式の解空間**:$A$を$m \\times n$行列とする．\n", "同次連立一次方程式$A\\boldsymbol{x} = \\boldsymbol{0}$の解全体\n", "$\\{\\boldsymbol{x} \\in \\boldsymbol{R}^n; A\\boldsymbol{x} = \\boldsymbol{0}\\}$\n", "は$ \\boldsymbol{R}^n$の部分空間をなす．\n", "これを$A\\boldsymbol{x} = \\boldsymbol{0}$の解空間という．\n", "\n", "* 部分空間の次元と基底\n", "$$\n", "{\\rm dim}\\{\\boldsymbol{x} \\in \\boldsymbol{R}^n; A\\boldsymbol{x} = \\boldsymbol{0}\\}\n", "= n -{\\rm rank}A\n", "$$\n", "であり，基本解は1組の基底である．" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 計量ベクトル空間\n", "* 内積を考えているとき$V$を**計量ベクトル空間**または**内積空間**という．\n", "* **内積**\n", "$V$を実数上の$n$次数ベクトル空間$\\boldsymbol{R}^n$またはその部分空間とする．\n", "$V$のベクトル$\\boldsymbol{a},\\boldsymbol{b}$に対して，実数$\\boldsymbol{a}\\cdot\\boldsymbol{b}$が定まり，つぎの(1)-(4)を満たすとき\n", "これを**内積**という．\n", " 1. $\\boldsymbol{a}\\cdot\\boldsymbol{b} = \\boldsymbol{b}\\cdot\\boldsymbol{a}$\n", " 1. $(\\boldsymbol{a}_1+\\boldsymbol{a}_2)\\cdot\\boldsymbol{b} =\n", "\\boldsymbol{a}_1 \\cdot \\boldsymbol{b} +\\boldsymbol{a}_2 \\cdot \\boldsymbol{b}$\n", " 1. $(\\lambda \\boldsymbol{a})\\cdot \\boldsymbol{b} =\n", "\\lambda (\\boldsymbol{a}\\cdot \\boldsymbol{b}) \\, (\\lambdaは実数)$\n", " 1. $\\boldsymbol{a} \\cdot\\boldsymbol{a} \\ge 0$ (等号は$\\boldsymbol{a}=0$に限る)\n", "\n", " \n", "\n", "* **自然な内積**\n", "$\\boldsymbol{R}^n$のベクトル$\\boldsymbol{a}=(a_1, a_2, \\ldots, a_n)$,\n", "$\\boldsymbol{b}=(b_1, b_2, \\ldots, b_n)$に対して\n", "$$\n", "\\boldsymbol{a} \\cdot\\boldsymbol{b} =\n", "a_1b_1 + a_2 b_2 + \\cdots + a_nb_n\n", "$$\n", "と定めると内積である．これを$\\boldsymbol{R}^n$の**自然な内積**という．\n", "\n", " * $\\boldsymbol{a},\\boldsymbol{b}$を列ベクトルとみなすと\n", "$$\n", "\\boldsymbol{a} \\cdot\\boldsymbol{b} = {^t \\boldsymbol{a}} \\boldsymbol{b}\n", "$$\n", "である．\n", "* **ベクトルの大きさ，長さ**\n", "計量ベクトル空間$V$のベクトル$\\boldsymbol{a}$にたいし\n", "$$\n", "\\left| \\boldsymbol{a} \\right | = \\sqrt{ \\boldsymbol{a} \\cdot \\boldsymbol{a} }\n", "$$\n", "を$\\boldsymbol{a}$の**大きさ**または**長さ**という\n", "\n", " * **長さの性質**\n", " 1. $\\left| \\boldsymbol{a} \\right | \\ge 0$ (等号は$ \\boldsymbol{a} = 0$のときに限る)\n", " 1. どんな実数$\\lambda$に対しても\n", "$\\left| \\lambda \\boldsymbol{a} \\right| = \\left| \\lambda \\right| \\left| \\boldsymbol{a} \\right|$\n", " 1. $\\left| \\boldsymbol{a} \\cdot \\boldsymbol{b} \\right| \\le \\left| \\boldsymbol{a} \\right| \\left| \\boldsymbol{b} \\right|$ (**シュヴァルツの不等式**)\n", " 1. $\\left| \\boldsymbol{a} + \\boldsymbol{b} \\right| \\le \\left| \\boldsymbol{a} \\right| + \\left| \\boldsymbol{b} \\right|$ (三角不等式)\n", "\n", "* **単位ベクトル，正規化**\n", " * $\\left| \\boldsymbol{e}\\right|=1$となるベクトル$\\boldsymbol{e}$を\n", "**単位ベクトル**という．\n", " * 零ベクトルでないベクトル$\\boldsymbol{a}$に対し，$\\pm \\frac{\\boldsymbol{a}}{\\left| \\boldsymbol{a} \\right|}$は単位ベクトルで，これを求める操作を**正規化する**という．\n", "* **交角**\n", "$$\n", "\\cos \\theta = \\frac{ \\boldsymbol{a} \\cdot \\boldsymbol{b} }{\\left| \\boldsymbol{a} \\right| \\left| \\boldsymbol{b} \\right|} \n", "\\, (0 \\le \\theta \\le \\pi)\n", "$$\n", "を満たす$\\theta $はただ一つであって，これを**交角**という．\n", "* **直交**\n", "$$\n", " \\boldsymbol{a} , \\boldsymbol{b} は互いに直交 \\iff \\boldsymbol{a} \\cdot \\boldsymbol{b} = 0\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 直交基底\n", "* **正規直交系**\n", "$m$個のベクトル$\\boldsymbol{a}_1, \\boldsymbol{a}_2, \\ldots, \\boldsymbol{a}_n$が\n", "$$\n", "\\boldsymbol{a}_i \\cdot \\boldsymbol{a}_j = \\delta_{ij}\\, (クロネッカーのデルタ)\n", "$$\n", "を満たすとき，これを**正規直交系**であるという\n", "\n", "* **正規直交基底**\n", "基底が正規直交系のとき**正規直交基底**という．\n", "\n", "* **グラム・シュミットの直交化法**\n", "\n", "* **直交補空間**\n", "$V$を$\\boldsymbol{R}^n$の部分空間とする．このとき\n", "$$\n", "V^{\\perp} = \\{\\boldsymbol{x}\\in\\boldsymbol{R}^n;すべての\\boldsymbol{y}\\in\\boldsymbol{V}に対して\\boldsymbol{x}\\cdot\\boldsymbol{y}=0\\}\n", "$$\n", "は$\\boldsymbol{R}^n$の部分空間で，これを$V$の**直交補空間**という．\n", "\n", " * $\\boldsymbol{R}^n = \\boldsymbol{V} \\oplus \\boldsymbol{V}^\\perp$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 直交補空間(orthogonal complement, perp)\n", "\n", "\n", "![magician genius](./figs/magician_genius.png)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# 紙\n", "\n", "![LA-II](./figs/LinearAlgebra-III.png)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# python\n", "\n", "上述の関係をpythonのmplot3dで確認．まず同次連立一次方程式$A\\boldsymbol{x} = \\boldsymbol{0}$の解は\n", "$$\n", "z = x/2 - y/2 \\, or\\\\\n", "x - y - 2z = 0\n", "$$\n", "であり，この時の面の法線ベクトルは，[1, -1, -2]である．\n", "これらのplotは次の通り．" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "d = 3\n", "x = np.arange(-d, d, 0.25)\n", "y = np.arange(-d, d, 0.25)\n", "X, Y = np.meshgrid(x, y)\n", "Z1 = f(X,Y)\n", "\n", "fig = plt.figure()\n", "\n", "plot3d = Axes3D(fig)\n", "plot3d.plot_surface(X,Y,Z1, color='y', alpha=0.1)\n", "\n", "soa = np.array([[0, 0, 0, 1, 1, 0], [0, 0, 0, 2, 0, 1]])\n", "X, Y, Z, U, V, W = zip(*soa)\n", "plot3d.quiver(X, Y, Z, U, V, W)\n", "\n", "soa = np.array([[0, 0, 0, 1, -1, 1], [1, 1, 0, 1, -1, 1]])\n", "# soa = np.array([[0, 0, 0, 1, -1, -2]])\n", "X, Y, Z, U, V, W = zip(*soa)\n", "plot3d.quiver(X, Y, Z, U, V, W, color='r')\n", "\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "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": "3.6.1" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": false }, "toc": { "colors": { "hover_highlight": "#DAA520", "navigate_num": "#000000", "navigate_text": "#333333", "running_highlight": "#FF0000", "selected_highlight": "#FFD700", "sidebar_border": "#EEEEEE", "wrapper_background": "#FFFFFF" }, "moveMenuLeft": true, "nav_menu": { "height": "12px", "width": "252px" }, "navigate_menu": true, "number_sections": true, "sideBar": true, "threshold": 4, "toc_cell": true, "toc_section_display": "block", "toc_window_display": false, "widenNotebook": false } }, "nbformat": 4, "nbformat_minor": 2 }