الجبر الخطي باستخدام NumPy

np.dot و@

import numpy as np

A = np.array([[1,2],[3,4]])
B = np.array([[5,6],[7,8]])

print(A @ B)
# [[19 22]
#  [43 50]]
print(np.dot(A, B))   # same result

الضرب على مستوى العناصر في مقابل ضرب المصفوفات

import numpy as np

A = np.array([[1,2],[3,4]])
B = np.array([[2,0],[1,3]])

print(A * B)    # [[ 2  0] [ 3 12]] — element-wise
print(A @ B)    # [[ 4  6] [10 12]] — matrix multiply

المنقول

import numpy as np

A = np.array([[1,2,3],[4,5,6]])
print(A.shape)    # (2, 3)
print(A.T.shape)  # (3, 2)
print(A.T)

np.linalg.inv وnp.linalg.det

import numpy as np

A = np.array([[2.,1.],[1.,3.]])
print(np.linalg.inv(A))
# [[ 0.6 -0.2]
#  [-0.2  0.4]]
print(np.linalg.det(A))   # 5.0

حل الأنظمة الخطية

import numpy as np

# Solve: 2x + y = 5; x + 3y = 10
A = np.array([[2.,1.],[1.,3.]])
b = np.array([5.,10.])
x = np.linalg.solve(A, b)
print(x)   # [1.  3.]   → x=1, y=3

القيم والمتجهات الذاتية

import numpy as np

A = np.array([[4., 2.], [1., 3.]])
vals, vecs = np.linalg.eig(A)
print("Eigenvalues:", vals)    # [5.  2.]
print("Eigenvectors:\n", vecs)

SVD: تحليل القيمة المفردة

import numpy as np

A = np.array([[1,2],[3,4],[5,6]], dtype=float)
U, s, Vt = np.linalg.svd(A, full_matrices=False)
print("Singular values:", s)
# Reconstruct: U @ np.diag(s) @ Vt ≈ A

np.linalg.norm

import numpy as np

v = np.array([3., 4.])
print(np.linalg.norm(v))           # 5.0  (Euclidean)
print(np.linalg.norm(v, ord=1))    # 7.0  (Manhattan)

A = np.eye(3)
print(np.linalg.norm(A, "fro"))    # 1.732 (Frobenius)

الجداء الاتجاهي والجداء الخارجي

import numpy as np

a = np.array([1,0,0])
b = np.array([0,1,0])
print(np.cross(a, b))   # [0 0 1]

print(np.outer([1,2,3], [4,5]))
# [[ 4  5]
#  [ 8 10]
#  [12 15]]

np.linalg.lstsq

import numpy as np

# Fit y = ax + b  to noisy data
X = np.column_stack([np.arange(5), np.ones(5)])
y = np.array([1.1, 2.0, 3.1, 3.9, 5.2])

coeffs, _, _, _ = np.linalg.lstsq(X, y, rcond=None)
print(f"slope={coeffs[0]:.2f}, intercept={coeffs[1]:.2f}")

np.einsum

import numpy as np

A = np.random.rand(3, 4)
B = np.random.rand(4, 5)

# Matrix multiplication via einsum:
C = np.einsum("ij,jk->ik", A, B)
print(C.shape)   # (3, 5)

# Trace:
M = np.eye(4)
print(np.einsum("ii->", M))   # 4.0

تحقّق سريع

مراجعة