数学 - Python - 線形代数学 - ベクトル空間 - 基底 - 実関数のなすベクトル空間、直線、逆数、指数関数、対数関数

学習環境

ラング線形代数学(上) (ちくま学現文庫)(S.ラング (著)、芹沢正三 (翻訳)、筑摩書房)の2章(ベクトル空間)、3(基底)、練習問題6の解答を求めてみる。

1. $\begin{array}{l} c_{1} t + c_{2} \frac{1}{t} = 0 \\ c_{1} t^{2} + c_{2} = 0 \\ 2 c_{1} t = 0 \\ c_{1} = 0 \\ c_{2} = 0 \end{array}$
  
  よって、 1次独立。
2. $\begin{array}{l} c_{1} e^{t} + c_{2} \log t = 0 \\ c_{1} e^{t} + \frac{c_{2}}{t} = 0 \\ c_{1} t e^{t} + c_{2} = 0 \\ c_{1} (e^{t} + t e^{t}) = 0 \\ c_{1} e^{t} (1 + t) = 0 \\ c_{1} = 0 \\ c_{2} = 0 \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, solve, plot, exp, log print('6.') t = symbols('t', real=True, positive=True) c1, c2 = symbols('c1, c2', imag=True) cs = [c1, c2] fs = [t, 1 / t] gs = [exp(t), log(t)] class LinearlyIndependentTestCase(TestCase): def test_a(self): inear_combination = sum([c * f for c, f in zip(cs, fs)]) self.assertEqual(solve(inear_combination, c1, c2), {c1: 0, c2: 0}) def test_b(self): inear_combination = sum([c * g for c, g in zip(cs, gs)]) self.assertEqual(solve(inear_combination, c1, c2), {c1: 0, c2: 0}) p = plot(*fs, *gs, (t, -5, 5), ylim=(-5, 5), legend=True, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'purple', 'pink', 'gray', 'skyblue', 'yellow'] for o, color in zip(p, colors): o.line_color = color p.show() p.save('sample6.png') if __name__ == '__main__': main()

% ./sample6.py -v 6. test_a (__main__.LinearlyIndependentTestCase) ... ok test_b (__main__.LinearlyIndependentTestCase) ... FAIL ====================================================================== FAIL: test_b (__main__.LinearlyIndependentTestCase) ---------------------------------------------------------------------- Traceback (most recent call last): File "./sample6.py", line 19, in test_b self.assertEqual(solve(inear_combination, c1, c2), {c1: 0, c2: 0}) AssertionError: [(-c2*exp(-t)*log(t), c2)] != {c1: 0, c2: 0} ---------------------------------------------------------------------- Ran 2 tests in 0.356s FAILED (failures=1) %

Kamimura's blog

2020年1月7日火曜日

数学 - Python - 線形代数学 - ベクトル空間 - 基底 - 実関数のなすベクトル空間、直線、逆数、指数関数、対数関数

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