数学 - Python - JavaScript - 線形代数学 - 線形写像 – 線形写像(ベクトル空間、平面、平行ではないベクトル、単位ベクトル、長方形の像、平行四辺形)

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12.を取り組んでみる。

12. 
    

    長方形は

    $\begin{array}{}{t}_1\left(0,1\right)+t_2\left(3,0\right)\\ 0\le \; <!--\; less-than\; or\; equal\; to\; -->t_1\le \; <!--\; less-than\; or\; equal\; to\; -->1,0\le \; <!--\; less-than\; or\; equal\; to\; -->t_2\le \; <!--\; less-than\; or\; equal\; to\; -->1\end{array}$

    T による像。

    $\begin{array}{}T\left(t_1\left(0,1\right)+t_2\left(3,0\right)\right)\\ =t_{1}T\left(0,1\right)+t_{2}T\left(3,0\right)\\ =t_{1}B+3t_{2}T\left(1,0\right)\\ =t_{1}B+3t_{2}A\\ =t_{1}B+t_2\left(3A\right)\end{array}$

    よって、 問題の長方形の T による像は、

    $3A,B$

    によって張られる平行四辺形。

コード(Emacs)

Python 3

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

A = Matrix([1, 2]).reshape(2, 1)
B = Matrix([2, 1]).reshape(2, 1)
E1 = Matrix([[1],
             [0]])
E2 = Matrix([[0],
             [1]])
T = Matrix([[1, 2],
            [2, 1]])
x, y = symbols('x, y')
p = Matrix([[x],
            [y]])


def f(x):
    return T * x

a = Matrix([0, 1]).reshape(2, 1)
b = Matrix([3, 0]).reshape(2, 1)
c = Matrix([0, 0]).reshape(2, 1)
d = Matrix([3, 1]).reshape(2, 1)

print(f(E1) == A, f(E2) == B)
for X in [A, B, p, a, b, c, d]:
    pprint(f(X))
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample12.py
True True
⎡5⎤
⎢ ⎥
⎣4⎦

⎡4⎤
⎢ ⎥
⎣5⎦

⎡x + 2⋅y⎤
⎢       ⎥
⎣2⋅x + y⎦

⎡2⎤
⎢ ⎥
⎣1⎦

⎡3⎤
⎢ ⎥
⎣6⎦

⎡0⎤
⎢ ⎥
⎣0⎦

⎡5⎤
⎢ ⎥
⎣7⎦

$

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="0.001">
<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">

<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="sample12.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 = (x, y) => x + 2 * y,
    fy = (x, y) => 2 * x + y;
                
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, 0, 0, 1, 'red'],
                 [0, 1, 3, 1, 'red'],
                 [3, 1, 3, 0, 'red'],
                 [3, 0, 0, 0, 'red'],
                 [fx(0, 0), fy(0, 0), fx(1, 0), fy(1, 0), 'green'],
                 [fx(1, 0), fy(1, 0), fx(1, 1), fy(1, 1), 'green'],
                 [fx(1, 1), fy(1, 1), fx(0, 1), fy(0, 1), 'green'],
                 [fx(0, 1), fy(0, 1), fx(0, 0), fy(0, 0), 'green']],
        fns = [];

    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 =