数学 - Python - 線形代数学 - 線形写像 – 写像(微分演算、積分、ベクトル、内積、座標関数、和、積)

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)
• 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
• 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)、1(写像)、練習問題1-7.を取り組んでみる。

1. 1. 
      $\cos x$
   2. 
      $e^x$
   3. 
      $\frac{1}{x}$
2. 1. 
      ${\displaystyle {\int }_0^x{e}^{t}dt}={\left[e^t\right]}_0^x=e^x-1$
   2. 
      $\begin{array}{l}{\displaystyle {\int }_0^x\frac{1}{1+t^2}dt}={\left[\arctan t\right]}_0^x\\ =\arctan x-\arctan 0\\ =\arctan x\end{array}$
   3. 
      $\begin{array}{l}{\displaystyle {\int }_0^x\cos tdt}={\left[\sin t\right]}_0^x\\ =\sin x-\sin 0\\ =\sin x\end{array}$
3. 1. 
      $\left(2,3,-1\right)·\left(1,2,-3\right)=2+6+3=11$
   2. 
      $\left(2,3,-1\right)·\left(-1,5,0\right)=-2+15=13$
   3. 
      $\left(2,3,-1\right)·\left(2,1,1\right)=4+3-1=6$
4. 
   $\begin{array}{l}F\left(1\right)=\left(e,1\right)\\ F\left(0\right)=\left(1,0\right)\\ F\left(-1\right)=\left(\frac{1}{e},-1\right)\end{array}$
5. 
   $\begin{array}{l}\left(F+G\right)\left(x\right)=\left(e^t+t,3t\right)\\ \left(F+G\right)\left(1\right)=\left(e+1,3\right)\\ \left(F+G\right)\left(2\right)=\left(e^2+2,6\right)\\ \left(F+G\right)\left(0\right)=\left(1,0\right)\end{array}$
6. 
   $\begin{array}{l}\left(2F\right)\left(x\right)=\left(2e^t,2t\right)\\ \left(2F\right)\left(0\right)=\left(2,0\right)\\ \left(\pi F\right)\left(x\right)=\left(\pi e^t,\pi t\right)\\ \left(\pi F\right)\left(1\right)=\left(\pi e,\pi \right)\end{array}$
7. 1. 
      $\begin{array}{l}F\left(X\right)=\left(1,1,0,-1\right)·\left(1,1,-1,3\right)+2\\ =1+1-3+2\\ =1\end{array}$
   2. 
      $\begin{array}{l}F(X)=(2,3,-1,1)·(1,1,-1,3)+2\hfill \\ =2+3+1+3+2\hfill \\ =11\hfill \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, sin, exp, log, cos, Derivative, Integral, Matrix

print('1.')
x = symbols('x')
for i, f in enumerate([sin, exp, log]):
    print('({0})'.format(chr(ord('a') + i)))
    g = Derivative(f(x), x, 1)
    pprint(g)
    pprint(g.doit())
    print()

print('2.')
t = symbols('t')
for i, f in enumerate([exp(t), 1 / (1 + t ** 2), cos(t)]):
    print('({0})'.format(chr(ord('a') + i)))
    g = Integral(f, (t, 0, x))
    pprint(g)
    pprint(g.doit())
    print()

print('3.')
A = Matrix([[2, 3, -1]])
for i, x in enumerate([[1, 2, -3], [-1, 5, 0], [2, 1, 1]]):
    print('({0})'.format(chr(ord('a') + i)))
    print(A.dot(x))
    print()

print('4.')
ft = Matrix([[exp(t), t]])
for i, t0 in enumerate([1, 0, -1]):
    print('({0})'.format(chr(ord('a') + i)))
    pprint(ft.subs({t: t0}))
    print()

print('5.')
gt = Matrix([[t, 2 * t]])
for i, t0 in enumerate([1, 2, 0]):
    print('({0})'.format(chr(ord('a') + i)))
    pprint((ft + gt).subs({t: t0}))
    print()

print('7.')
A = [1, 1, -1, 3]
for i, x0 in enumerate([(1, 1, 0, -1), (2, 3, -1, 1)]):
    print('({0})'.format(chr(ord('a') + i)))
    X = Matrix([x0])
    print(X.dot(A) + 2)

入出力結果(Terminal, IPython)

$ ./sample1.py
1.
(a)
d         
──(sin(x))
dx        
cos(x)

(b)
d ⎛ x⎞
──⎝ℯ ⎠
dx    
 x
ℯ 

(c)
d         
──(log(x))
dx        
1
─
x

2.
(a)
x      
⌠      
⎮  t   
⎮ ℯ  dt
⌡      
0      
 x    
ℯ  - 1

(b)
x          
⌠          
⎮   1      
⎮ ────── dt
⎮  2       
⎮ t  + 1   
⌡          
0          
atan(x)

(c)
x          
⌠          
⎮ cos(t) dt
⌡          
0          
sin(x)

3.
(a)
11

(b)
13

(c)
6

4.
(a)
[ℯ  1]

(b)
[1  0]

(c)
⎡ -1    ⎤
⎣ℯ    -1⎦

5.
(a)
[1 + ℯ  3]

(b)
⎡     2   ⎤
⎣2 + ℯ   6⎦

(c)
[1  0]

7.
(a)
1
(b)
11
$