数学 - Python - JavaScript - 線形代数学 - 線形写像 – 正値スカラー積(連続実関数のなすベクトル空間、積分、基本性質)

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7章(スカラー積と直交性)、2(正値スカラー積)、練習問題3.を取り組んでみる。

3. 
   

   SP 1について。

   $\begin{array}{}<f,g>\\ ={\int \; <!--\; integral\; -->}_0^{1}f\left(t\right)g\left(t\right)\mathrm{dt}\\ ={\int \; <!--\; integral\; -->}_0^{1}g\left(t\right)f\left(t\right)\mathrm{dt}\\ =<g,f>\end{array}$

   SP 2について。

   $\begin{array}{}<f,g+h>\\ ={\int \; <!--\; integral\; -->}_0^{1}f\left(t\right)\left(g\left(t\right)+h\left(t\right)\right)\mathrm{dt}\\ ={\int \; <!--\; integral\; -->}_0^1\left(f\left(t\right)g\left(t\right)+f\left(t\right)h\left(t\right)\right)\mathrm{dt}\\ ={\int \; <!--\; integral\; -->}_0^{1}f\left(t\right)g\left(t\right)\mathrm{dt}+{\int \; <!--\; integral\; -->}_0^{1}f\left(t\right)h\left(t\right)\mathrm{dt}\\ =⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->f,g⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->+⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->f,h⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->\end{array}$

   SP 3について。

   $\begin{array}{}<xf,g>\\ ={\int \; <!--\; integral\; -->}_0^{1}xf\left(t\right)g\left(t\right)\mathrm{dt}\\ =x{\int \; <!--\; integral\; -->}_0^{1}f\left(t\right)g\left(t\right)\mathrm{dt}\\ =x<f,g>\\ <f,xg>\\ ={\int \; <!--\; integral\; -->}_0^{1}f\left(t\right)xg\left(t\right)\mathrm{dt}\\ =x{\int \; <!--\; integral\; -->}_0^{1}f\left(t\right)g\left(t\right)\mathrm{dt}\\ =x<f,g>\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sin, exp, Integral, plot

print('3.')
t, x = symbols('t, x')
f = t ** 2
g = exp(t)
h = sin(t)


def sp(f, g):
    return Integral(f * g, (t, 0, 1))


ts = [(sp(f, g), sp(g, f)),
      (sp(f, g + h), sp(f, g) + sp(f, h)),
      (sp(x * f, g), x * sp(f, g)),
      (sp(f, x * g), x * sp(f, g))]

for s in ts:
    for u in list(s) + [s[0].doit().simplify() == s[1].doit().simplify()]:
        pprint(u)
        print()
    print()

p = plot(*[k.subs({x: 5}) for k in [f, g, h, f * g, f * (g + h), x * f * g]],
         ylim=(-5, 5), legend=True, show=False)
colors = ['red', 'green', 'blue', 'orange', 'purple', 'brown']
for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample3.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample3.py
3.
1         
⌠         
⎮  2  t   
⎮ t ⋅ℯ  dt
⌡         
0         

1         
⌠         
⎮  2  t   
⎮ t ⋅ℯ  dt
⌡         
0         

True


1                    
⌠                    
⎮  2 ⎛ t         ⎞   
⎮ t ⋅⎝ℯ  + sin(t)⎠ dt
⌡                    
0                    

1                1         
⌠                ⌠         
⎮  2             ⎮  2  t   
⎮ t ⋅sin(t) dt + ⎮ t ⋅ℯ  dt
⌡                ⌡         
0                0         

True


1           
⌠           
⎮  2    t   
⎮ t ⋅x⋅ℯ  dt
⌡           
0           

  1         
  ⌠         
  ⎮  2  t   
x⋅⎮ t ⋅ℯ  dt
  ⌡         
  0         

True


1           
⌠           
⎮  2    t   
⎮ t ⋅x⋅ℯ  dt
⌡           
0           

  1         
  ⌠         
  ⎮  2  t   
x⋅⎮ t ⋅ℯ  dt
  ⌡         
  0         

True


$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<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">

<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="sample3.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    input_a1 = document.querySelector('#a1'),
    input_b1 = document.querySelector('#b1'),
    input_a2 = document.querySelector('#a2'),
    input_b2 = document.querySelector('#b2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => x ** 2,
    g = (x) => Math.exp(x),
    h = (x) => Math.sin(x),
    fns = [[f, 'red'],
           [g, 'green'],
           [h, 'blue'],
           [(x) => f(x) * g(x), 'orange'],
           [(x) => f(x) * (g(x) + h(x)), 'purple'],
           [(x) => 5 * f(x) * g(x), 'brown']];

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, y1, 0, y2, 'pink'],
                 [1, y1, 1, y2, 'skyblue']];

    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 =