線形代数における「基底の変換」と「線形写像（線形変換）」の関係

参考リンクまとめ

Pythonコード

# -------------------------------
# ライブラリのインポート / Import libraries
# -------------------------------
import numpy as np

# -------------------------------
# 線形空間 V → W の線形写像 A / Linear map A (from V to W)
# -------------------------------
A = np.array([
    [2, 1],
    [0, 3],
    [1, -1]
])  # 3x2 行列, V: R^2 → W: R^3

# -------------------------------
# V空間の基底変換行列 P（2x2）/ Change of basis for V
# -------------------------------
P = np.array([
    [1, 1],
    [0, 2]
])  # 新しい基底 → 標準基底への変換（右から作用）

# -------------------------------
# W空間の基底変換行列 Q（3x3）/ Change of basis for W
# -------------------------------
Q = np.array([
    [1, 0, 1],
    [0, 2, 0],
    [0, 0, 1]
])  # 新しい基底 → 標準基底への変換（右から作用）

# -------------------------------
# 成分ベクトル x（新しいV基底における）/ x under new basis of V
# -------------------------------
x_new = np.array([
    [2],
    [1]
])

# -------------------------------
# B = Q⁻¹ A P による新しい基底での線形写像 / Linear map under new basis
# -------------------------------
Q_inv = np.linalg.inv(Q)
B = Q_inv @ A @ P  # 新しい基底での線形写像行列

# -------------------------------
# y_new = B x_new / 新しい基底での出力成分
# -------------------------------
y_new = B @ x_new

# -------------------------------
# 標準基底での結果と比較 / Compare with original space
# -------------------------------
# x の標準基底表現
x_std = P @ x_new

# 標準基底での y = A x
y_std = A @ x_std

# 新しい基底での y_new → 標準基底に戻す: Q y_new
y_std_from_new = Q @ y_new

# -------------------------------
# 結果表示 / Print results
# -------------------------------
np.set_printoptions(precision=2, suppress=True)

print("Matrix A (standard basis):\n", A)
print("\nMatrix P (change of basis for V):\n", P)
print("\nMatrix Q (change of basis for W):\n", Q)
print("\nMatrix B (transformed linear map):\n", B)

print("\nInput x (new basis):\n", x_new)
print("Input x (standard basis):\n", x_std)

print("\nOutput y = A x (standard basis):\n", y_std)
print("Output y = B x (new basis):\n", y_new)
print("y (from new basis, converted to standard):\n", y_std_from_new)

結果
Matrix A (standard basis):
[[ 2 1]
[ 0 3]
[ 1 -1]]

Matrix P (change of basis for V):
[[1 1]
[0 2]]

Matrix Q (change of basis for W):
[[1 0 1]
[0 2 0]
[0 0 1]]

Matrix B (transformed linear map):
[[ 1. 5.]
[ 0. 3.]
[ 1. -1.]]

Input x (new basis):
[[2]
[1]]
Input x (standard basis):
[[3]
[2]]

Output y = A x (standard basis):
[[8]
[6]
[1]]
Output y = B x (new basis):
[[7.]
[3.]
[1.]]
y (from new basis, converted to standard):
[[8.]
[6.]
[1.]]