4.1 張成與線性獨立：用圖形看空間#

import numpy as np
import matplotlib.pyplot as plt
from linalg_viz import plot_vectors

np.random.seed(0)

def scan_span(vectors, n=400, lim=2.0):
    """對每個向量的係數均勻掃描，畫出所有線性組合的終點。"""
    coefs = np.random.uniform(-lim, lim, size=(n, len(vectors)))
    pts = coefs @ np.array(vectors)          # (n, k) @ (k, dim) -> (n, dim)
    return pts

# 情況 A：單一向量 -> 直線
v = np.array([2.0, 1.0])
pts = scan_span([v])
plt.figure(figsize=(5, 5))
plt.scatter(pts[:, 0], pts[:, 1], s=4, alpha=0.4)
plot_vectors([v], color="red")
plt.gca().set_aspect("equal"); plt.title("span{v}：一條直線")
plt.show()

# 情況 B：兩個不共線向量 -> 平面
u, w = np.array([2.0, 0.0]), np.array([1.0, 2.0])
pts = scan_span([u, w])
plt.figure(figsize=(5, 5))
plt.scatter(pts[:, 0], pts[:, 1], s=4, alpha=0.4)
plot_vectors([u, w], color=["red", "green"])
plt.gca().set_aspect("equal"); plt.title("span{u, w}：整個平面")
plt.show()

# 情況 C：兩個共線向量 -> 仍是直線
u2 = np.array([1.0, 1.0])
w2 = np.array([2.0, 2.0])    # 是 u2 的兩倍，沒有新地盤
pts = scan_span([u2, w2])
plt.figure(figsize=(5, 5))
plt.scatter(pts[:, 0], pts[:, 1], s=4, alpha=0.4)
plot_vectors([u2, w2], color=["red", "green"])
plt.gca().set_aspect("equal"); plt.title("span{u, 2u}：還是那條直線")
plt.show()

def is_independent(*vectors):
    """用秩判斷一組向量是否線性獨立。"""
    A = np.array(vectors).T          # 每個向量一行
    return np.linalg.matrix_rank(A) == A.shape[1]

print(is_independent([2, 0], [1, 2]))          # True ：撐起平面
print(is_independent([1, 1], [2, 2]))          # False：共線
print(is_independent([1, 0, 0], [0, 1, 0], [1, 1, 0]))  # False：共面（都在 z=0 平面）
print(is_independent([1, 0, 0], [0, 1, 0], [0, 0, 1]))  # True ：標準基

def classify_span_3d(v1, v2, v3):
    A = np.column_stack([v1, v2, v3])
    r = np.linalg.matrix_rank(A)
    return {1: "直線（三向量共線）",
            2: "平面（三向量共面）",
            3: "整個 R^3（線性獨立）"}[r]

cases = {
    "共線":   ([1, 2, 3], [2, 4, 6], [-1, -2, -3]),
    "共面":   ([1, 0, 0], [0, 1, 0], [1, 1, 0]),   # 全在 z=0 平面
    "獨立":   ([1, 0, 0], [0, 1, 0], [0, 0, 1]),
}
for name, (a, b, c) in cases.items():
    print(f"{name}: {classify_span_3d(np.array(a, float), np.array(b, float), np.array(c, float))}")

# 隨機 10000 組：三個隨機 3D 向量幾乎總是獨立！
from collections import Counter
counts = Counter()
for _ in range(10000):
    vs = [np.random.randn(3) for _ in range(3)]
    counts[classify_span_3d(*vs)] += 1
print(counts)

# 3D 共面情況的視覺化：畫出三向量與它們所在的平面
from mpl_toolkits.mplot3d import Axes3D  # noqa: F401

vs = [np.array([1., 0., 0.]), np.array([0., 1., 0.]), np.array([1., 1., 0.])]
fig = plt.figure(figsize=(6, 6))
ax = fig.add_subplot(projection="3d")
s, t = np.meshgrid(np.linspace(-2, 2, 10), np.linspace(-2, 2, 10))
ax.plot_surface(s, t, 0 * s, alpha=0.15, color="gray")     # z=0 平面
for v in vs:
    ax.quiver(0, 0, 0, *v, color=np.random.rand(3), arrow_length_ratio=0.1)
ax.set_box_aspect([1, 1, 1]); ax.set_title("三個共面向量")
plt.show()