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Grade 12 Math Lab: Analysis & Inference FREE

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Null Space & Linear Algebra Foundations

65 min Grade 12 Math Lab: Analysis & Inference

Learning Objectives

  • Find null space of a matrix
  • Understand rank-nullity theorem
  • Solve homogeneous systems

Key Concepts

Null Space

Set of all vectors x where Ax = 0

Rank-Nullity

rank(A) + nullity(A) = number of columns

Theory

**Null Space** (kernel) of A: all solutions to Ax = 0 **To find:** 1. Row reduce A to RREF 2. Express free variables as parameters 3. Write solution as linear combination of basis vectors **Rank-Nullity Theorem:** rank(A) + nullity(A) = n (# columns) **Trivial null space:** Only x = 0 → columns are linearly independent.

Examples

Null space of [[1,2],[2,4]]

Solution: RREF: [[1,2],[0,0]]. x₁=-2x₂. Null space = span{⟨-2,1⟩}