2017年11月3日金曜日

学習環境

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4.を取り組んでみる。


  1. x 1 ( a,b )+ x 2 ( c,d )=0 a x 1 +c x 2 =0 b x 1 +d x 2 =0 ab x 1 +bc x 2 =0 ab x 1 +ad x 2 =0 ( adbc ) x 2 =0 ad x 1 +cd x 2 =0 bc x 1 +cd x 2 =0 ( adbc ) x 1 =0

    ad - bc = 0のとき、x2は0ではなくても成り立つ。

    よって、問題の平面上の2つのベクトル(a, b)、(b, c)は1次従属である。

    また、ad - bc が0ではないときはx1 = 0 かつ x2 = 0なので、問題の平面上の2つのベクトル(a, b)、(b, c)は1次独立である。

コード(Emacs)

Python 3

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

print('4.')
a, b, c, d, x1, x2 = symbols('a b c d x1 x2')
v1 = Matrix([a, b])
v2 = Matrix([c, d])
eq = x1 * v1 + x2 * v2
pprint(solve(eq))

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
⎡⎧   -c⋅x₂      -d⋅x₂ ⎫                ⎤
⎢⎨a: ──────, b: ──────⎬, {x₁: 0, x₂: 0}⎥
⎣⎩     x₁         x₁  ⎭                ⎦
$

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="a0">a = </label>
<input id="a0" type="number" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" value="3">
<label for="c0">c = </label>
<input id="c0" type="number" value="-2">
<label for="d0">d = </label>
<input id="d0" type="number" value="-3">
<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">


<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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    input_c0 = document.querySelector('#c0'),
    input_d0 = document.querySelector('#d0'),
    input_c1 = document.querySelector('#c1'),
    input_c2 = document.querySelector('#c2'),
    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0, input_c0, input_d0, input_c1, input_c2],
    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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value),
        c0 = parseFloat(input_c0.value),
        d0 = parseFloat(input_d0.value),
        c1 = parseFloat(input_c1.value),
        c2 = parseFloat(input_c2.value);

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

    let points = [],
        lines = [[0, 0, a0, b0, 'red'],
                 [0, 0, c0, d0, 'green'],
                 [0, 0, c1 * a0 + c2 * c0, c1 * b0 + c2 * d0, 'blue']],
        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();









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