Rank, Nullity and Column Space: Understanding What a Matrix Preserves and Loses
A matrix transforms input vectors into output vectors. Some independent directions survive, some are combined, and some may disappear completely into the zero vector. Span, dimension, column space, rank, null space and nullity describe this process precisely. input space → matrix transformation → column space 1. Span: all reachable combinations Given vectors v₁, v₂, …, vₖ, their span is the set of every vector that can be made by scaling and adding them: Span{v₁, v₂, …, vₖ} = {c₁v₁ + c₂v₂ + ⋯ + cₖvₖ : c₁, c₂, …, cₖ ∈ ℝ}. One non-zero direction spans a line through the origin. Two independent directions span a plane through the origin. Three independent directions in ℝ³ span all of ℝ³. 2. Independence and dimension Vectors are linearly independent when none of them can be constructed from the others. Each independent vector contributes a genuinely new direction. A dependent vector contributes no new...