数学 - Python - JavaScript - 線形代数学 - R^n におけるベクトル – ベクトルのノルム(ベクトルの長さ、スカラー積(内積)、平方根)

ラング線形代数学(上)
原著

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

ラング線形代数学(上)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の1章(R^n におけるベクトル)、3(ベクトルのノルム)、練習問題1、2.を取り組んでみる。

1. 1. 
      $\sqrt{4+1}=\sqrt{5}$
   2. 
      $\sqrt{1+9}=\sqrt{10}$
   3. 
      $\sqrt{4+1+25}=\sqrt{30}$
   4. 
      $\sqrt{1+4+9}=\sqrt{14}$
   5. 
      $\sqrt{{\pi }^2+9+1}=\sqrt{{\pi }^2+10}$
   6. 
      $\sqrt{225+4+16}=\sqrt{245}=7\sqrt{5}$
2. 1. 
      $\sqrt{1+1}=\sqrt{2}$
   2. 
      $\sqrt{16}=4$
   3. 
      $\sqrt{1+1+1}=\sqrt{3}$
   4. 
      $\sqrt{1+9+16}=\sqrt{26}$
   5. 
      $\sqrt{4{\pi }^2+9+49}=\sqrt{4{\pi }^2+58}$
   6. 
      $\sqrt{{\pi }^2+9+1}=\sqrt{{\pi }^2+10}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, Matrix, pi

A = [(2, -1),
     (-1, 3),
     (2, -1, 5),
     (-1, -2, 3),
     (pi, 3, -1), (15, -2, 4)]

B = [(-1, 1),
     (0, 4),
     (-1, 1, 1),
     (-1, 3, -4),
     (2 * pi, -3, 7),
     (pi, 3, -1)]

for i, X in enumerate([A, B], 1):
    print(f'{i}.')
    for j, t in enumerate(X, 1):
        M = Matrix(t)
        for x in [M.T, M.norm()]:
            pprint(x)
        print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
1.
[2  -1]
√5

[-1  3]
√10

[2  -1  5]
√30

[-1  -2  3]
√14

[π  3  -1]
   _________
  ╱  2      
╲╱  π  + 10 

[15  -2  4]
7⋅√5

2.
[-1  1]
√2

[0  4]
4

[-1  1  1]
√3

[-1  3  -4]
√26

[2⋅π  -3  7]
   ___________
  ╱    2      
╲╱  4⋅π  + 58 

[π  3  -1]
   _________
  ╱  2      
╲╱  π  + 10 

$

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="dx">dx = </label>
<input id="dx" 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">

<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="sample1.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_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

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

    let points = [],
        lines = [[0, 0, 2, -1, 'red'],
                 [0, 0, -1, 1, 'green'],
                 [0, 0, -1, 3, 'blue'],
                 [0, 0, 0, 4, 'orange']],
        fns = [],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                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 = dx =
x1 = x2 =
y1 = y2 =