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University Math Lab: Systems & Structures FREE

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Linear Transformations

60 min University Math Lab: Systems & Structures

Learning Objectives

  • Represent transformations as matrices
  • Compose transformations
  • Find image and kernel

Key Concepts

Linear Transformation

T(au + bv) = aT(u) + bT(v)

Transformation Matrix

Matrix A where T(x) = Ax

Theory

**Linear transformation** T: ℝⁿ → ℝᵐ satisfies: - T(u + v) = T(u) + T(v) - T(cv) = cT(v) **Matrix representation:** T(x) = Ax **Standard transformations in ℝ²:** - Rotation by θ: [[cosθ, -sinθ], [sinθ, cosθ]] - Reflection over x-axis: [[1, 0], [0, -1]] - Scaling by k: [[k, 0], [0, k]] - Shear: [[1, k], [0, 1]] **Composition:** T₂(T₁(x)) = (A₂A₁)x

Examples

Rotate (1,0) by 90°

Solution: [[0,-1],[1,0]]·[1,0]ᵀ = (0, 1)