#範例14-8：證明A 的核空間和 A 的列空間彼此正交(垂直)
from sympy import *
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_nullsapce = M.nullspace()
M_rowspace = M.rowspace()
print('線性轉換後有壓縮空間 = 核空間向量集合 = null space=\n', M_nullsapce)
print('輸出列空間的向量之線性組和= column space= \n', M_rowspace)
n1, n2, n3 = M.nullspace()
r1, r2, r3 = M.rowspace()
print('n1 = ', n1)
print('r1 = ', r1)
print('n1.dot(r1)=核空間和列空間彼此正交(內積=0) = ', n1.dot(r1))