数学 - Python - JavaScript - 線型代数 - ベクトル空間 - 1次従属と1次独立(R^2(二次元)、1次従属、方程式)

線型代数入門

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

線型代数入門(松坂 和夫(著)、岩波書店)の第2章(ベクトル空間)、6(1次従属と1次独立)、問2.を取り組んでみる。

2. 
   $\begin{array}{l}\left(1+x\right)·\left(1+x\right)-2x·2x=0\\ x^2+2x+1-4x^2=0\\ 3x^2-2x-1=0\\ \left(x-1\right)\left(3x+1\right)=0\\ x=-\frac{1}{3},1\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, solve

x = symbols('x')

eq = (1 + x) ** 2 - (2 * x) ** 2

pprint(eq)
pprint(solve(eq, dict=True))

入出力結果(Terminal, IPython)

$ ./sample2.py 
     2          2
- 4⋅x  + (x + 1) 
[{x: -1/3}, {x: 1}]
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dΘ">dΘ = </label>
<input id="dΘ" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="x0">x = </label>
<input id="x0" type="number" value="1">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample2.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dΘ = document.querySelector('#dΘ'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dΘ, input_x1, input_x2, input_y1, input_y2,
              input_x0],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dΘ = parseFloat(input_dΘ.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        x0 = parseFloat(input_x0.value);

    if (r === 0 || dΘ === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [[0, 0, 1 + x0, 2 * x0, 'red'],
                 [0, 0, 2 * x0, 1 + x0, 'green']],
        fns = [],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let Θ = 0; Θ <= 2 * Math.PI; Θ += dΘ) {
                let r = f(Θ),
                    x = r * Math.cos(Θ),
                    y = r * Math.sin(Θ);

                points.push([x, y, color]);
            }
        });
    
    fns2
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dΘ =
x1 = x2 =
y1 = y2 =
x =