数学 - Python - JavaScript - 線形代数学 - 線形写像 – 正値スカラー積(定数関数、1次関数、2次関数、連続実関数のなすベクトル空間、正規直交基底)

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5.を取り組んでみる。

5. 
   

   直交基底。

   $\begin{array}{}1\\ t-\frac{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->t,1⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->1,1⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}·\; <!--\; middle\; dot\; -->1\\ =t-\frac{{\int \; <!--\; integral\; -->}_0^{1}t\mathrm{dt}}{{\int \; <!--\; integral\; -->}_0^{1}1\mathrm{dt}}·\; <!--\; middle\; dot\; -->1\\ =t-\frac{1}{2}\\ t^2-\frac{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->t^2,t-\frac{1}{2}⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->t-\frac{1}{2},t-\frac{1}{2}⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}\left(t-\frac{1}{2}\right)-\frac{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->t^2,1⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->1,1⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}·\; <!--\; middle\; dot\; -->1\\ =t^2-\frac{{\int \; <!--\; integral\; -->}_0^1\left(t^3-\frac{1}{2}{t}^2\right)\mathrm{dt}}{{\int \; <!--\; integral\; -->}_0^1\left(t^2-t+\frac{1}{4}\right)\mathrm{dt}}\left(t-\frac{1}{2}\right)-\frac{1}{3}\\ =t^2-\frac{\frac{1}{4}-\frac{1}{6}}{\frac{1}{3}-\frac{1}{2}+\frac{1}{4}}\left(t-\frac{1}{2}\right)-\frac{1}{3}\\ =t^2-\frac{3-2}{4-6+3}\left(t-\frac{1}{2}\right)-\frac{1}{3}\\ =t^2-t+\frac{1}{6}\end{array}$

   正規直交基底。

   $\begin{array}{}\frac{1}{\sqrt{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->1,1⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}}=1\\ \frac{t-\frac{1}{2}}{\sqrt{⟨\; <!--\; mathematical\; left\; angle\; bracket\; -->t-\frac{1}{2},t-\frac{1}{2}⟩\; <!--\; mathematical\; right\; angle\; bracket\; -->}}\\ =\sqrt{12}\left(t-\frac{1}{2}\right)\\ =\sqrt{3}\left(2t-1\right)\\ \frac{t^2-t+\frac{1}{6}}{\sqrt{{\int \; <!--\; integral\; -->}_0^1\left(t^4-2t^3+\frac{4}{3}{t}^2-\frac{1}{3}t+\frac{1}{36}\right)\mathrm{dt}}}\\ =\frac{t^2-t+\frac{1}{6}}{\sqrt{\frac{1}{5}-\frac{1}{2}+\frac{4}{9}-\frac{1}{6}+\frac{1}{36}}}\\ =\sqrt{\frac{180}{36-90+80-30+5}}\left(t^2-t+\frac{1}{6}\right)\\ =6\sqrt{5}\left(t^2-t+\frac{1}{6}\right)\\ =\sqrt{5}\left(6t^2-6t+1\right)\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, Integral, Rational, plot

print('4.')
t = symbols('t')


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


def norm(f):
    return sqrt(sp(f, f))


f = 1
g = t
h = t ** 2

v1 = f / sp(f, f)
v2 = g - sp(g, v1) / sp(v1, v1) * v1
v3 = h + sum([-sp(h, v0) / sp(v0, v0) * v0 for v0 in [v2, v1]])
for v in [v1, v2, v3]:
    for s in [v, (v / norm(v)).expand()]:
        pprint(s)
        print()
    print()
p = plot(f, g, h, v1, v2, v3, ylim=(-10, 10), legend=True, show=False)
colors = ['red', 'green', 'blue', 'orange', 'brown', 'purple']
for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample5.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
4.
1

1


t - 1/2

2⋅√3⋅t - √3


 2       1
t  - t + ─
         6

      2              
6⋅√5⋅t  - 6⋅√5⋅t + √5


$

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="sample5.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) => 1,
    g = (x) => x,
    h = (x) => x ** 2,
    v2 = (x) => Math.sqrt(3) * (2 * x - 1),
    v3 = (x) => Math.sqrt(5) * (6 * x ** 2 - 6 * x + 1)
    fns = [[f, 'red'],
           [g, 'green'],
           [h, 'blue'],
           [v2, 'orange'],
           [v3, '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 = [];

    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 =