数学 - Python - JavaScript - 線形代数学 - ベクトル空間 – 基底(三角関数(基底)、整数倍、1次独立)

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

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2章(ベクトル空間)、3(基底)、練習問題11.を取り組んでみる。

$\begin{array}{l} x_{1} \sin t + \dots + x_{n} \sin (n t) = 0 \\ x_{1} \sin t \sin (k t) + \dots + x_{k} \sin^{2} (k t) + \dots + x_{n} \sin (n t) \sin (k t) = 0 \\ \int_{- π}^{π} (x_{1} \sin t \sin (k t) + \dots + x_{k} \sin^{2} (k t) + \dots + x_{n} \sin (n t) \sin (k t)) dt \\ = \int_{- π}^{π} 0 dt \end{array}$

前問10より、

$\begin{array}{l} \int_{- π}^{π} x_{k} \sin^{2} (k t) dt = 0 \\ x_{k} = 0 \\ (k = 1, 2, \dots, n) \end{array}$

よって、問題の関数は実数上で1次独立である。（証明終）

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sin, solve
import random

print('11.')
t = symbols('t', real=True)
for n in range(1, 6):
    s = symbols([f'x{i}' for i in range(1, n + 1)])
    eq = sum([x * y for x, y in
              zip(s, [sin(i * t) for i in range(1, n + 1)])])
    for o in [eq, solve(eq, s)]:
        pprint(o)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample11.py
11.
x₁⋅sin(t)

[0]


x₁⋅sin(t) + x₂⋅sin(2⋅t)

[{x₁: -2⋅x₂⋅cos(t)}]


x₁⋅sin(t) + x₂⋅sin(2⋅t) + x₃⋅sin(3⋅t)

⎡⎧                   x₃⋅sin(3⋅t)⎫⎤
⎢⎨x₁: -2⋅x₂⋅cos(t) - ───────────⎬⎥
⎣⎩                      sin(t)  ⎭⎦


x₁⋅sin(t) + x₂⋅sin(2⋅t) + x₃⋅sin(3⋅t) + x₄⋅sin(4⋅t)

⎡⎧    -(x₂⋅sin(2⋅t) + x₃⋅sin(3⋅t) + x₄⋅sin(4⋅t)) ⎫⎤
⎢⎨x₁: ───────────────────────────────────────────⎬⎥
⎣⎩                       sin(t)                  ⎭⎦


x₁⋅sin(t) + x₂⋅sin(2⋅t) + x₃⋅sin(3⋅t) + x₄⋅sin(4⋅t) + x₅⋅sin(5⋅t)

⎡⎧    -(x₂⋅sin(2⋅t) + x₃⋅sin(3⋅t) + x₄⋅sin(4⋅t) + x₅⋅sin(5⋅t)) ⎫⎤
⎢⎨x₁: ─────────────────────────────────────────────────────────⎬⎥
⎣⎩                              sin(t)                         ⎭⎦


$

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">
<br>
<label for="c1">c1 = </label>
<input id="c1" type="number" value="1">
<label for="c2">c2 = </label>
<input id="c2" type="number" value="1">
<label for="c3">c3 = </label>
<input id="c3" type="number" value="1">
<label for="c4">c4 = </label>
<input id="c4" type="number" value="1">
<label for="c5">c5 = </label>
<input id="c5" type="number" value="1">
<label for="c6">c6 = </label>
<input id="c6" type="number" value="1">
<label for="c7">c7 = </label>
<input id="c7" type="number" value="1">
<label for="c8">c8 = </label>
<input id="c8" type="number" value="1">
<label for="c9">c9 = </label>
<input id="c9" type="number" value="1">
<label for="c10">c10 = </label>
<input id="c10" type="number" value="1">

<button id="draw0">draw</button>
<button id="clear0">cplear</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="sample11.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_c1 = document.querySelector('#c1'),
    input_c2 = document.querySelector('#c2'),
    input_c3 = document.querySelector('#c3'),
    input_c4 = document.querySelector('#c4'),
    input_c5 = document.querySelector('#c5'),
    input_c6 = document.querySelector('#c6'),
    input_c7 = document.querySelector('#c7'),
    input_c8 = document.querySelector('#c8'),
    input_c9 = document.querySelector('#c9'),
    input_c10 = document.querySelector('#c10'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_c1, input_c2, input_c3, input_c4, input_c5,
              input_c6, input_c7, input_c8, input_c9, input_c10],
    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),
        c1 = parseFloat(input_c1.value),
        c2 = parseFloat(input_c2.value),
        c3 = parseFloat(input_c3.value),
        c4 = parseFloat(input_c4.value),
        c5 = parseFloat(input_c5.value),
        c6 = parseFloat(input_c6.value),
        c7 = parseFloat(input_c7.value),
        c8 = parseFloat(input_c8.value),
        c9 = parseFloat(input_c9.value),
        c10 = parseFloat(input_c10.value);

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

    let points = [],
        lines = [],
        f = (x) =>
        [c1, c2, c3, c4, c5, c6, c7, c8, c9, c10]
        .map((c, i) => c * Math.sin((i + 1) * x))
        .reduce((a, b) => a + b),
        fns = [[f, 'green']],
        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 =
c1 = c2 = c3 = c4 = c5 = c6 = c7 = c8 = c9 = c10 =

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2017年11月10日金曜日

数学 - Python - JavaScript - 線形代数学 - ベクトル空間 – 基底(三角関数(基底)、整数倍、1次独立)

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