IndianTechnoEra: Linear Algebra For AIML - Chapter 5 Linear Transformations

Chapter 5 — Linear Transformations

Linear transformations are operations that map vectors from one space to another, often using matrices. They are fundamental in AI & ML for manipulating data, performing feature engineering, and transforming images in computer vision.

5.1 What is a Linear Transformation?

A linear transformation is a function T that maps vectors v in one vector space to vectors in another vector space while preserving vector addition and scalar multiplication:

T(u + v) = T(u) + T(v)
T(α * v) = α * T(v)

In matrix form, applying a transformation is usually done by multiplying a matrix A with a vector v:
v_transformed = A × v

5.2 Types of Linear Transformations

Common linear transformations in 2D or 3D:

• Rotation: Rotates a vector around the origin.
• Scaling: Changes the size of vectors in one or more directions.
• Shearing: Slants the shape along an axis.
• Translation: Shifts vectors by adding a constant vector (note: strictly speaking, translation is affine, not linear).

Importance in AI/ML: Transformations allow us to manipulate features, normalize data, rotate images, or adjust coordinate systems — essential for preprocessing, augmentation, and geometric operations in models.

5.3 Matrix Representation

Every linear transformation can be represented as a matrix multiplication:

A = [[2, 0],
     [0, 3]]  # scaling
v = [1, 1]
v_transformed = A @ v  # [2, 3]

Here, A scales x by 2 and y by 3.

5.4 Quick NumPy Examples (Practical)

import numpy as np

# Define a 2D vector
v = np.array([1, 1])

# Scaling transformation
A = np.array([[2, 0],
              [0, 3]])
v_scaled = A @ v
print("Scaled vector:", v_scaled)

# Rotation 90 degrees counterclockwise
theta = np.pi/2
R = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta),  np.cos(theta)]])
v_rotated = R @ v
print("Rotated vector:", v_rotated)

# Shearing transformation
S = np.array([[1, 1],
              [0, 1]])
v_sheared = S @ v
print("Sheared vector:", v_sheared)

5.5 Geometric Intuition

- Scaling stretches or compresses vectors along axes.
- Rotation spins vectors around the origin.
- Shearing slants shapes, like turning a rectangle into a parallelogram.
- Visualizing transformations helps understand how data is manipulated in feature space or images.

5.6 AI/ML Use Cases (Why Linear Transformations Matter)

• Computer Vision: Image rotations, scaling, and shearing for data augmentation improve model robustness.
• Feature Engineering: Transforming features (e.g., scaling, PCA) to optimize model performance.
• Neural Networks: Each linear layer in a neural network is a learned linear transformation.
• Graphics & Robotics: Coordinate transformations for 3D modeling and movement planning.

5.7 Exercises

1. Scale the vector [3, 4] by 2x in x and 0.5x in y using a matrix.
2. Rotate vector [1, 0] by 180° using a rotation matrix.
3. Shear the vector [2,1] along x-axis with shear factor 1.
4. Verify that a rotation preserves the vector magnitude (||v||).

Answers / Hints

1. Scaling matrix: [[2,0],[0,0.5]], result: [6, 2]
2. Rotation matrix 180°: [[-1,0],[0,-1]], result: [-1,0]
3. Shear matrix: [[1,1],[0,1]], result: [3,1]
4. Magnitude ||[cosθ, sinθ]|| = 1, remains unchanged under rotation.

5.8 Practice Projects / Mini Tasks

• Take a small grayscale image (matrix of pixels) and apply scaling, rotation, and shear using NumPy.
• Implement a simple data augmentation pipeline for a dataset of 2D points.
• Visualize original vs transformed vectors using matplotlib to see effects of linear transformations.

5.9 Further Reading & Videos

• 3Blue1Brown — Essence of Linear Algebra (YouTube series on transformations).
• NumPy documentation — matrix operations, rotation/scaling matrices.
• Linear Algebra textbooks — chapters on linear transformations and eigenvectors.

Next chapter: Eigenvalues & Eigenvectors — learn how to compute them, their geometric meaning, and their importance in PCA, SVD, and ML models.