5) Linear Transformations

Linear Transformations

Linear Transformations

1. What is a Linear Transformation?

A linear transformation is a function that maps vectors from one vector space to another while preserving vector addition and scalar multiplication.

Real-World Meaning:

  • Image Processing: Transforming an image (scaling, rotating, shearing).
  • Computer Graphics: Converting 3D objects into 2D projections.
  • Physics: Changing reference frames in mechanics or relativity.

2. Properties of Linear Transformations

A function T: V \to W is a linear transformation if it satisfies:

  • Additivity: T(u + v) = T(u) + T(v)
  • Homogeneity: T(c v) = c T(v)

Real-World Meaning:

  • Resizing an image: A transformation that doubles the size of an image maintains proportionality.
  • Rotating a figure: The shape stays the same, only its orientation changes.

3. Matrix Representation of Linear Transformations

Every linear transformation can be represented as a matrix multiplication:

T(v) = A v

Real-World Meaning:

  • Animation & Video Effects: Transformations like zooming, skewing, or rotating are done using matrices.
  • Machine Learning: Weight matrices in neural networks apply linear transformations to input data.

4. Common Linear Transformations in 2D & 3D

  • Scaling: Enlarges or shrinks an object.
  • Rotation: Turns an object around a point.
  • Reflection: Flips an object over a line.
  • Shear: Slants the shape while keeping areas unchanged.

5. Kernel and Image of a Transformation

  • Kernel (Null Space): The set of all vectors mapped to zero.
  • Image (Column Space): The set of all vectors that can be reached by the transformation.

Real-World Meaning:

  • Kernel in Data Science: Finding dependency among features (removing redundant information).
  • Image in Graphics: Determining what part of a 3D object is visible in a 2D scene.

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