The Null Space: Everything A Sends to Zero

Lesson notes

Key points

• The null space of A is the set of all vectors x satisfying the homogeneous equation Ax equals zero.
• Row reduce A, mark the free variables, and each free variable gives one basis vector of the null space.
• The rank-nullity theorem links the dimensions: rank plus nullity equals the number of columns n.
• If Ax equals b has one solution p, every solution is p plus any null space vector.

Definitions

Null space N(A)

All vectors x with Ax = 0; always contains the zero vector and is a subspace.

Takeaway: Null space = all solutions of Ax = 0; its size is n minus rank.

Null spaceFree variablesRank-nullityGeneral solution