数学 - 線型代数 - JavaScript - 複素数、複素ベクトル空間(複素平面、三角不等式、等号)

線型代数入門

学習環境/開発環境

• 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, Google Chrome...)用JavaScript Library: MathJax
• OS X El Capitan - Apple (OS)
• Emacs (Text Editor)
• JavaScript (プログラミング言語)
• kjs-math-number (JavaScript Library)
• Safari(Web browser)

線型代数入門 (松坂 和夫(著)、岩波書店)の第4章(複素数、複素ベクトル空間)、2(複素平面)、問4.を取り組んでみる。

問4.

$\begin{array}{l}\alpha =0\\ \left|0+\beta \right|=\left|\beta \right|\\ \left|0\right|+\left|\beta \right|=\left|\beta \right|\\ \\ \beta =c\alpha \wedge c\ge 0\\ \left|\alpha +c\alpha \right|=\left(c+1\right)\left|\alpha \right|\\ \left|\alpha \right|+\left|c\alpha \right|=\left|\alpha \right|+c\left|\alpha \right|=\left(1+c\right)\left|\alpha \right|\\ \\ {\left(\left|\alpha \right|+\left|\beta \right|\right)}^2={\left|\alpha \right|}^2+2\left|\alpha \right|\left|\beta \right|+{\left|\beta \right|}^2\\ =\alpha \overline{\alpha }+2\left|\alpha \right|\left|\beta \right|+\beta \overline{\beta }\\ 2\left|\alpha \right|\left|\beta \right|=2\mathrm{Re}\left(\alpha \overline{\beta }\right)\\ \left|\alpha \right|\left|\beta \right|=\mathrm{Re}\left(\alpha \overline{\beta }\right)\\ \alpha =0\\ \left|\alpha \right|\left|\beta \right|=0\\ \mathrm{Re}\left(\alpha \overline{\beta }\right)=0\\ \alpha \ne 0\\ \alpha =a+bi,\beta =c+di,\sqrt{a^2+b^2}\ne 0\\ \overline{\beta }=c-di\\ \alpha \overline{\beta }=\left(a+bi\right)\left(c-di\right)\\ =\left(ac+bd\right)-\left(ad+bd\right)i\\ \mathrm{Re}\left(\alpha \overline{\beta }\right)=ac+bd\\ \left|\alpha \right|\left|\beta \right|=\sqrt{a^2+b^2}\sqrt{c^2+d^2}\\ a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2-a^2{c}^2-b^2{d}^2-2abcd\\ =a^2{d}^2+b^2{c}^2-2abcd\\ ={\left(ad-bc\right)}^2\\ ad=bc\end{array}$

number.js で確認。

JavaScript

コード(Emacs)

(function () {
    'use strict';
    var div_output = document.querySelector('#output0'),
        button_calc = document.querySelector('#calc0'),        
        nl = '<br>',
        display;

    display = function () {
        var a,
            b,
            a,
            b,
            c = Math.floor(Math.random() * 100) + 1,
            left,
            right,
            output = '';
        
        a = 0;
        b = new Complex(Math.floor(Math.random() * 100) + 1,
                        Math.floor(Math.random() * 100) + 1);
        left = a.add(b).abs();
        right = a.abs().add(b.abs());
        output += 
            'a = <math><mn>' + a + '</mn></math>, b = <math>' + b + nl +
            '左辺: <math><mn>' + left + '</mn></math>' + nl +
            '右辺: <math><mn>' + right + '</mn></math>' + nl +
            left.isEqual(right) + nl + 
            '差: ' + left.sub(right).abs() + nl;
        
        a = new Complex(Math.floor(Math.random() * 100),
                        Math.floor(Math.random() * 100));
        b = c.mul(a);
        output += 
            'a = <math>' + a + '</math>, b = <math>' + b + nl +
            '左辺: <math><mn>' + left + '</mn></math>' + nl +
            '右辺: <math><mn>' + right + '</mn></math>' + nl +
            left.isEqual(right) + nl + 
            '差: ' + left.sub(right).abs() + nl;
        div_output.innerHTML = output;
    };
    
    button_calc.onclick = display;
    
    display();
}());