Null Space & Nullity

Linear Algebra for Machine Learning · Axiom Academy

The null space (or kernel) of a matrix A is the set of all vectors x such that Ax = 0. It consists of all solutions to the homogeneous equation. Finding the Null Space: Worked Example Step 3: Identify free variables From the row echelon form, x₃ is free. Express x₁, x₂ in terms of x₃: The null space is all scalar multiples of this vector. Dimension = 1. The null space always contains the zero vector If only trivial solution, null space is 0 with dimension 0 Dimension of null space = nullity = n - rank(A) Non-trivial solutions exist when rank(A) < n A is invertible ⟺ null space = 0 A is singular ⟺ null space contains non-zero vectors rank(A) = n ⟺ null space is trivial For a non-homogeneous system Ax = b: If x₀ is a particular solution and x_h is in the null space of A, then: gives all solutions to Ax = b.

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