%matplotlib inline
import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
我們要判斷 $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n$ 是否線性獨立, 只要令一個矩陣
$$A = \begin{bmatrix}\mathbf{x}_1 \mathbf{x}_2 \cdots \mathbf{x}_n\end{bmatrix}$$注意向量是以行 (column) 向量表示。求了 $A$ 的 reduced row echelon from (rref), 有 pivot (軸元) 相對應的行向量, 就是線性獨立的向量。
At = sp.Matrix([[2, 1, -3], [1, -2, 4], [17, -4, 2]])
A = At.transpose()
A
A.rref()
(Matrix([ [1, 0, 6], [0, 1, 5], [0, 0, 0]]), (0, 1))
所以 $(17,-4,2) \in span(𝑆)$
At = sp.Matrix([[2, 1, -3], [1, -2, 4], [17, -4, 5]])
A = At.transpose()
A
A.rref()[0]
設 $V = \mathbb{R}^4$, 且
$$W = \mathrm{span}((1,1,3,2), (2, 2, 6, 4), (1,1,0,1))$$A = sp.Matrix([[1,1,3,2], [2,2,6,4], [1,1,0,1]]).transpose()
A
A.rref()[0]
A = sp.Matrix([[1,1,3,2], [1,1,0,1], [1,0,0,0], [0,1,0,0],
[0,0,1,0],[0,0,0,1]]).transpose()
A
A.rref()
(Matrix([ [1, 0, 0, 0, 1/3, 0], [0, 1, 0, 0, -2/3, 1], [0, 0, 1, 0, 1/3, -1], [0, 0, 0, 1, 1/3, -1]]), (0, 1, 2, 3))