数学 - Python - JavaScript - 線型代数 - 線型写像 - 行列の積(原点のまわりの回転の行列、三角関数、加法定理)

線型代数入門

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

線型代数入門(松坂 和夫(著)、岩波書店)の第3章(線型写像)、7(行列の積)、問題4-(a)、(b).を取り組んでみる。

4. 1. 
      

      原点のまわりの角β回転して、さらに角α回転するのは、原点のまわりの角α + β回転することと同じ。よって、問題のことが成り立つ。

   2. 
      

      角αの回転について。

      $\begin{array}{l}{\phi }_{\alpha }\left(e_1\right)=\left(\begin{array}{c}\cos \alpha \\ \sin \alpha \end{array}\right)\\ {\phi }_{\alpha }\left(e_2\right)=\left(\begin{array}{c}-\sin \alpha \\ \cos \alpha \end{array}\right)\\ \\ {\phi }_{\alpha }\left(\left(\begin{array}{c}x\\ y\end{array}\right)\right)\\ ={\phi }_{\alpha }\left(x{e}_1+y{e}_2\right)\\ =x{\phi }_{\alpha }\left(e_1\right)+y{\phi }_{\alpha }\left(e_2\right)\\ =x\left(\begin{array}{c}\cos \alpha \\ \sin \alpha \end{array}\right)+y\left(\begin{array}{c}-\sin \alpha \\ \cos \alpha \end{array}\right)\\ =\left(\begin{array}{c}x\cos \alpha \\ x\sin \alpha \end{array}\right)+\left(\begin{array}{c}-y\sin \alpha \\ y\cos \alpha \end{array}\right)\\ =\left(\begin{array}{c}x\cos \alpha -y\sin \alpha \\ x\sin \alpha +y\cos \alpha \end{array}\right)\end{array}$

      βついても同様に求める。

      ${\phi }_{\beta }\left(\left(\begin{array}{c}x\\ y\end{array}\right)\right)=\left(\begin{array}{c}x\cos \beta -y\sin \beta \\ x\sin \beta +y\cos \beta \end{array}\right)$

      合成関数について。

      $\begin{array}{l}\left({\phi }_{\alpha }\circ {\phi }_{\beta }\right)\left(\left(\begin{array}{c}x\\ y\end{array}\right)\right)\\ ={\phi }_{\alpha }\left({\phi }_{\beta }\left(\left(\begin{array}{c}x\\ y\end{array}\right)\right)\right)\\ ={\phi }_{\alpha }\left(\left(\begin{array}{c}x\cos \beta -y\sin \beta \\ x\sin \beta +y\cos \beta \end{array}\right)\right)\\ =\left(\begin{array}{c}\left(x\cos \beta -y\sin \beta \right)\cos \alpha -\left(x\sin \beta +y\cos \beta \right)\sin \alpha \\ \left(x\cos \beta -y\sin \beta \right)\sin \alpha +\left(x\sin \beta +y\cos \beta \right)\cos \alpha \end{array}\right)\\ =\left(\begin{array}{c}x\left(\cos \alpha \cos \beta -\sin \alpha \sin \beta \right)-y\left(\sin \alpha \cos \beta +\cos \alpha \sin \beta \right)\\ x\left(\sin \alpha \cos \beta +\cos \alpha \sin \beta \right)+y\left(\cos \alpha \cos \beta -\sin \alpha \sin \beta \right)\end{array}\right)\\ =\left(\begin{array}{cc}\cos \alpha \cos \beta -\sin \alpha \sin \beta & \sin \alpha \cos \beta +\cos \alpha \sin \beta \\ \sin \alpha \cos \beta +\cos \alpha \sin \beta & \cos \alpha \cos \beta -\sin \alpha \sin \beta \end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\end{array}$

      原点のまわりの角α + βの回転について。

      $\begin{array}{l}{\phi }_{\alpha +\beta }\left(\left(\begin{array}{c}x\\ y\end{array}\right)\right)\\ =\left(\begin{array}{c}x\cos \left(\alpha +\beta \right)-y\sin \left(\alpha +\beta \right)\\ x\sin \left(\alpha +\beta \right)+y\cos \left(\alpha +\beta \right)\end{array}\right)\\ =\left(\begin{array}{cc}\cos \left(\alpha +\beta \right)& -\sin \left(\alpha +\beta \right)\\ \sin \left(\alpha +\beta \right)& \cos \left(\alpha +\beta \right)\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)\end{array}$

      よって(a)より、次のことが成り立つ。

      $\begin{array}{l}\left(\begin{array}{cc}\cos \left(\alpha +\beta \right)& -\sin \left(\alpha +\beta \right)\\ \sin \left(\alpha +\beta \right)& \cos \left(\alpha +\beta \right)\end{array}\right)=\left(\begin{array}{cc}\cos \alpha \cos \beta -\sin \alpha \sin \beta & -\left(\sin \alpha \cos \beta +\cos \alpha \sin \beta \right)\\ \sin \alpha \cos \beta +\cos \alpha \sin \beta & \cos \alpha \cos \beta -\sin \alpha \sin \beta \end{array}\right)\\ \\ \cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\ \sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \end{array}$

      加法定理の証明完了。

コード(Emacs)

Python 3

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

print('4.')
a, b = symbols('a b')
f = lambda a: Matrix([[cos(a), -sin(a)],
                      [sin(a), cos(a)]])

x = f(a + b)
y = f(a) * f(b)

for t in [x, y]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
⎡cos(a + b)  -sin(a + b)⎤
⎢                       ⎥
⎣sin(a + b)  cos(a + b) ⎦

⎡-sin(a)⋅sin(b) + cos(a)⋅cos(b)  -sin(a)⋅cos(b) - sin(b)⋅cos(a)⎤
⎢                                                              ⎥
⎣sin(a)⋅cos(b) + sin(b)⋅cos(a)   -sin(a)⋅sin(b) + cos(a)⋅cos(b)⎦

$

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="x0">x0 = </label>
<input id="x0" type="number" value="5">
<label for="y0">y0 = </label>
<input id="y0" type="number" value="0">

<label for="a0">a0 = </label>
<input id="a0" type="number" step="0.01" value="1">
<label for="b0">b0 = </label>
<input id="b0" type="number" step="0.01" 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="sample4.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_x0 = document.querySelector('#x0'),
    input_y0 = document.querySelector('#y0'),
    input_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_x0, input_y0, input_a0, input_b0],
    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 fx = (a, x, y) => x * Math.cos(a) - y * Math.sin(a),
    fy = (a, x, y) => x * Math.sin(a) + y * Math.cos(a);

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),
        x0 = parseFloat(input_x0.value),
        y0 = parseFloat(input_y0.value),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);
        

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

    let points = [],
        lines = [[0, 0, x0, y0, 'red'],
                 [0, 0, fx(a0, x0, y0), fy(a0, x0, y0), 'green'],
                 [0, 0, fx(b0, x0, y0), fy(b0, x0, y0), 'blue'],
                 [0, 0, fx(a0 + b0, x0, y0), fy(a0 + b0, x0, y0), '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 =
x0 = y0 = a0 = b0 =