W Code - Pattern-Based DSA Learning Platform

Bhanu Bisht

Linear Algebra

The backbone of every machine learning algorithm

Topics

Level 0

Foundation

🇮🇳 Indian Context

Delhi NCR housing, document similarity for Hindi/English, and cricket analytics!

Vectors & Vector Spaces

Foundation

Python Implementation

import numpy as np

print("=" * 55)
print("VECTORS IN ML")
print("=" * 55)

# Define vectors
u = np.array([3, 4])
v = np.array([1, 2])

# Operations
print(f"\nu = {u}")
print(f"v = {v}")
print(f"\nu + v = {u + v}")
print(f"3 × u = {3 * u}")

# Dot Product
dot = np.dot(u, v)
print(f"\nu · v = {dot}")
print(f"Manual: {u[0]*v[0]} + {u[1]*v[1]} = {u[0]*v[0] + u[1]*v[1]}")

# Magnitude
mag_u = np.linalg.norm(u)
mag_v = np.linalg.norm(v)
print(f"\n‖u‖ = √({u[0]}² + {u[1]}²) = √{u[0]**2 + u[1]**2} = {mag_u:.4f}")

# Cosine Similarity
cos_sim = dot / (mag_u * mag_v)
angle = np.degrees(np.arccos(cos_sim))
print(f"\nCosine Similarity = {cos_sim:.4f}")
print(f"Angle between u and v = {angle:.1f}°")

# ML Example: Linear Prediction
print("\n" + "=" * 55)
print("ML: PREDICTION = DOT PRODUCT")
print("=" * 55)

features = np.array([1500, 3, 5])  # sqft, bedrooms, age
weights = np.array([0.05, 15, -2])  # learned weights
bias = 40

prediction = np.dot(features, weights) + bias
print(f"\nFeatures: Area={features[0]}sqft, BHK={features[1]}, Age={features[2]}yr")
print(f"Weights:  {weights}")
print(f"Bias:     {bias}")
print(f"Price = w·x + b = {np.dot(features, weights)} + {bias} = ₹{prediction} Lakhs")

# Cosine Similarity for NLP
print("\n📊 Indian Example: Document Similarity")
doc1 = np.array([3, 1, 0, 2])  # word counts: [cricket, movie, food, india]
doc2 = np.array([2, 0, 0, 3])
doc3 = np.array([0, 2, 3, 0])
sim_12 = np.dot(doc1, doc2) / (np.linalg.norm(doc1) * np.linalg.norm(doc2))
sim_13 = np.dot(doc1, doc3) / (np.linalg.norm(doc1) * np.linalg.norm(doc3))
print(f"Sports article vs India article: {sim_12:.4f}")
print(f"Sports article vs Food article:  {sim_13:.4f}")
print(f"Higher similarity = more related content")

Key Takeaways

Vectors are ordered lists of numbers — every data point is a vector

Dot product u·v = Σuᵢvᵢ computes predictions (ŷ = w·x + b)

Cosine similarity measures angle between vectors (used in NLP, recommendations)

Magnitude ‖v‖ = √(Σvᵢ²) gives vector length

Vector spaces must satisfy closure, zero vector, and inverse properties

import numpy as np print("=" * 55) print("VECTORS IN ML") print("=" * 55) # Define vectors u = np.array([3, 4]) v = np.array([1, 2]) # Operations print(f"\nu = {u}") print(f"v = {v}") print(f"\nu + v = {u + v}") print(f"3 × u = {3 * u}") # Dot Product dot = np.dot(u, v) print(f"\nu · v = {dot}") print(f"Manual: {u[0]*v[0]} + {u[1]*v[1]} = {u[0]*v[0] + u[1]*v[1]}") # Magnitude mag_u = np.linalg.norm(u) mag_v = np.linalg.norm(v) print(f"\n‖u‖ = √({u[0]}² + {u[1]}²) = √{u[0]**2 + u[1]**2} = {mag_u:.4f}") # Cosine Similarity cos_sim = dot / (mag_u * mag_v) angle = np.degrees(np.arccos(cos_sim)) print(f"\nCosine Similarity = {cos_sim:.4f}") print(f"Angle between u and v = {angle:.1f}°") # ML Example: Linear Prediction print("\n" + "=" * 55) print("ML: PREDICTION = DOT PRODUCT") print("=" * 55) features = np.array([1500, 3, 5]) # sqft, bedrooms, age weights = np.array([0.05, 15, -2]) # learned weights bias = 40 prediction = np.dot(features, weights) + bias print(f"\nFeatures: Area={features[0]}sqft, BHK={features[1]}, Age={features[2]}yr") print(f"Weights: {weights}") print(f"Bias: {bias}") print(f"Price = w·x + b = {np.dot(features, weights)} + {bias} = ₹{prediction} Lakhs") # Cosine Similarity for NLP print("\n📊 Indian Example: Document Similarity") doc1 = np.array([3, 1, 0, 2]) # word counts: [cricket, movie, food, india] doc2 = np.array([2, 0, 0, 3]) doc3 = np.array([0, 2, 3, 0]) sim_12 = np.dot(doc1, doc2) / (np.linalg.norm(doc1) * np.linalg.norm(doc2)) sim_13 = np.dot(doc1, doc3) / (np.linalg.norm(doc1) * np.linalg.norm(doc3)) print(f"Sports article vs India article: {sim_12:.4f}") print(f"Sports article vs Food article: {sim_13:.4f}") print(f"Higher similarity = more related content")