🔄 Linear Transformation

Last Updated: Jan 2026

A linear transformation is a rule that:

  • Takes a vector as input
  • Produces a new vector as output
  • Uses a matrix to do the transformation

🗣 Hinglish Tip: Linear transformation = vector ko matrix se pass karke badalna

Basic Idea

Input Vector  →  Matrix  →  Output Vector
  • Vector = data
  • Matrix = rule
  • Output = transformed data

Mathematical Representation

T(v) = A × v

Where:

  • v = input vector
  • A = transformation matrix
  • T(v) = transformed vector

Scaling

Input Vector

v = [2, 3]

Transformation Matrix

A = [2  0
     0  2]

Output

A × v = [4, 6]

Meaning:

  • Size changed
  • Direction stayed same

🗣 Hinglish Tip: Scaling = vector ko bada ya chhota karna

Types of Linear Transformations

Scaling

Changes size of vector.

Used in:

  • Feature normalization
  • Image resizing

Rotation

Changes direction of vector.

Used in:

  • Games
  • Animations
  • Graphics engines

Translation

Changes position of vector.

Used in:

  • Image alignment
  • Graphics rendering
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