Linear Algebra with NumPy
np.dot, np.linalg.inv, np.linalg.eig, np.linalg.solve for systems of equations.
Linear Algebra with NumPy is a free Learn AI with Python lesson on CoddyKit — lesson 4 of 4. You can read the complete lesson below for free — then practise it hands-on in the browser with a built-in code editor and a 24/7 AI tutor. It is part of the Learn AI with Python learning path, one of 4 lessons in the course, and your progress syncs across the web and the CoddyKit app.
Linear Algebra in NumPy
NumPy provides matrix operations through the @ operator and the np.linalg module, the foundation of nearly every machine learning algorithm.
Matrix Multiplication with @
The @ operator performs true matrix multiplication. Inner dimensions must match: (m,k) @ (k,n) gives (m,n).
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]]@ vs * (Important!)
* is element-wise, @ is matrix multiply. Confusing them is a classic bug.
print(A * B) # element-wise: [[ 5 12]
# [21 32]]
print(A @ B) # matrix product (different result)np.dot
np.dot is the function form. For 2D arrays it equals @; for 1D arrays it computes the dot product (a scalar).
u = np.array([1, 2, 3])
v = np.array([4, 5, 6])
print(np.dot(u, v)) # 1*4 + 2*5 + 3*6 = 32Matrix Inverse
np.linalg.inv(A) returns the inverse, the matrix that yields the identity when multiplied by A. Only square, non-singular matrices have one.
A = np.array([[4.0, 7.0], [2.0, 6.0]])
Ainv = np.linalg.inv(A)
print(np.round(A @ Ainv, 6)) # identityDeterminant
np.linalg.det(A) gives the determinant. A determinant of 0 means the matrix is singular and cannot be inverted.
print(np.linalg.det(A)) # 10.0 (nonzero -> invertible)Solving Linear Systems
To solve A x = b, prefer np.linalg.solve(A, b) over computing the inverse. It is faster and more numerically stable.
A = np.array([[3.0, 1.0], [1.0, 2.0]])
b = np.array([9.0, 8.0])
x = np.linalg.solve(A, b)
print(x) # [2. 3.]Why solve over inv
Computing inv(A) @ b wastes work and amplifies rounding error. solve uses LU decomposition directly. Reach for solve in real code.
# Avoid: x = np.linalg.inv(A) @ b
# Prefer: x = np.linalg.solve(A, b)Eigenvalues and Eigenvectors
np.linalg.eig(A) returns eigenvalues and eigenvectors. These underpin PCA, stability analysis, and more.
A = np.array([[2.0, 0.0], [0.0, 3.0]])
vals, vecs = np.linalg.eig(A)
print(vals) # [2. 3.]Vector and Matrix Norms
np.linalg.norm measures magnitude. By default it is the Euclidean (L2) norm for vectors, used everywhere from distances to regularization.
v = np.array([3.0, 4.0])
print(np.linalg.norm(v)) # 5.0 (sqrt(9+16))
print(np.linalg.norm(v, 1)) # 7.0 (L1 norm)Transpose
A.T transposes a matrix, swapping rows and columns. It appears constantly in formulas like the normal equations.
A = np.array([[1, 2, 3], [4, 5, 6]])
print(A.T)
# [[1 4]
# [2 5]
# [3 6]]Quick Check
Test your linear algebra recall.
Recap
Linear algebra toolkit:
@/np.dotfor matrix and dot products (not*)np.linalg.inv,np.linalg.detnp.linalg.solve(A, b)beats invertingnp.linalg.eigfor eigen-decompositionnp.linalg.normfor magnitudes,A.Tto transpose
Frequently asked questions
Is the “Linear Algebra with NumPy” lesson free?
Yes — the full text of “Linear Algebra with NumPy” is free to read here on the web, and the Learn AI with Python course includes 4 lessons in total. To practise it interactively (a built-in code editor and a 24/7 AI tutor) and unlock the rest of the Learn AI with Python course, upgrade to CoddyKit PRO.
What will I learn in “Linear Algebra with NumPy”?
np.dot, np.linalg.inv, np.linalg.eig, np.linalg.solve for systems of equations. You practise Learn AI with Python with hands-on code you run directly in the browser, and a 24/7 AI tutor answers your questions as you work through the lesson.
Do I need any experience to start Learn AI with Python?
No prior experience is required. Learn AI with Python on CoddyKit is structured for beginners through advanced learners; this is — lesson 4 of 4, so you can start here or from the beginning and move at your own pace.
How long does the “Linear Algebra with NumPy” lesson take?
Most CoddyKit lessons take about 5–10 minutes. Each one is bite-sized and interactive, so you make steady progress and pick up exactly where you left off across the web and the app.
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