数学 - Python - JavaScript - 線形代数学 - 線形写像 – 線形写像(ベクトル空間、実数、n次元)

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 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4章(線形写像)、2(線形写像)、練習問題6.を取り組んでみる。

6. 
   
   $\begin{array}{}u,v\in \; <!--\; element\; of\; -->V\\ F\left(u+v\right)\\ =\left(f\left(u+v\right),g\left(u+v\right)\right)\\ =\left(f\left(u\right)+f\left(v\right),g\left(u\right)+g\left(v\right)\right)\\ =\left(f\left(u\right),f\left(u\right)\right)+\left(g\left(v\right),g\left(v\right)\right)\\ =f\left(u\right)+f\left(v\right)\\ c\in \; <!--\; element\; of\; -->K\\ F\left(cv\right)\\ =\left(f\left(cv\right),g\left(cv\right)\right)\\ =\left(cf\left(v\right),\mathrm{cg}\left(v\right)\right)\\ =c\left(f\left(v\right),g\left(v\right)\right)\\ =cf\left(v\right)\end{array}$

   一般化。

   $\begin{array}{}F\left(v\right)=\left(f_1\left(v\right),\dots \; <!--\; horizontal\; ellipsis\; -->,f_n\left(v\right)\right)\\ F\left(u+v\right)=F\left(u\right)+F\left(v\right)\\ F\left(cv\right)=cF\left(v\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Matrix

x, a, b, c = symbols('x, a, b, c')
f = 2 * x
g = 3 * x
F = Matrix([f, g])

l1 = F.subs({x: a + b})
r1 = F.subs({x: a}) + F.subs({x: b})
l2 = F.subs({x: c * a})
r2 = c * F.subs({x: a})
for t in [F, l1, r1, l1 == r1, l2, r2, l2 == r2]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
⎡2⋅x⎤
⎢   ⎥
⎣3⋅x⎦

⎡2⋅a + 2⋅b⎤
⎢         ⎥
⎣3⋅a + 3⋅b⎦

⎡2⋅a + 2⋅b⎤
⎢         ⎥
⎣3⋅a + 3⋅b⎦

True

⎡2⋅a⋅c⎤
⎢     ⎥
⎣3⋅a⋅c⎦

⎡2⋅a⋅c⎤
⎢     ⎥
⎣3⋅a⋅c⎦

True

$

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="0.001" value="0.005">
<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="c0">c = </label>
<input id="c0" type="number" value="2">

<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="sample6.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'),
    input_c0 = document.querySelector('#c0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_c0],
    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 f = (x) => 2 * x;
                
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),
        c0 = parseFloat(input_c0.value);

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

    let points = [],
        g = (x) => f(c0 * x),
        h = (x) => c0 * f(x),
        lines = [],
        fns = [[f, 'red'],
               [g, 'blue'],
               [h, 'green']];

    fns
        .forEach((o) => {
            let [f, color] = o;

            for (let x0 = x1; x0 <= x2; x0 += dx) {
                points.push([x0, f(x0), 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);

    p(fns.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
c =