#範例14-2：無限多解系統的通解（齊次特解x0+非齊次解xh）
#比較sympy模組，numpy模組的解法)
#（1）sympy求解
from sympy import *
x1,x2,x3,x4,x5,x6 = symbols('x1 x2 x3 x4 x5 x6')
#注意：M矩陣的最右邊=y=轉換後座標
M = Matrix([
    [1,3,-2,0,2,0,0],
    [2,6,-5,-2,4,-3,-1],
    [0,0,5,10,0,15,5],
    [2,6,0,8,4,18,6]
    ])
M_simple = M.rref()
print('顯示高斯消去法簡化的梯形矩陣的echelon form =\n', M_simple)
ans = solve_linear_system(M, x1,x2,x3,x4,x5,x6)
print('用sympy解聯立方程式 = ', ans)

M = Matrix([
    [1,3,-2,0,2,0],
    [2,6,-5,-2,4,-3],
    [0,0,5,10,0,15],
    [2,6,0,8,4,18]
    ])
M_rank = M.rank()
M_dim = M.shape[1]
M_nullity = M_dim - M_rank

print('輸入空間維度 = input dimension of M=', M_dim)
print('輸出空間維度 = rank(M)=', M_rank)
print('核空間維度 = 被轉換壓縮的空間維度 = nullity=', M_nullity)
M_nullsapce = M.nullspace()
print('線性轉換後有壓縮空間 = 核空間向量集合 = null space=\n', M_nullsapce)


#（2）numpy求解
import numpy as np
A = np.array([
    [1,3,-2,0,2,0],
    [2,6,-5,-2,4,-3],
    [0,0,5,10,0,15],
    [2,6,0,8,4,18]
    ])
Y = np.array([
    [0],[-1],[5],[6]
    ])
print('用numpy解聯立方程式，X=（無法處理無限多組解）') 
X = np.linalg.solve(A, Y)
print(X)