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The null space nul(A) is the set of all solutions to Ax = 0, found by Gaussian elimination. It is a subspace and equals the kernel of the linear transformation x ↦ Ax. Nullity(A) is the dimension of nul(A), equal to the number of free variables in RREF.
The null space of a matrix A, written nul(A), is the set of all vectors x such that Ax = 0. You already know from Gaussian elimination how to solve this: row-reduce A to RREF and read off the solutions. What the null space concept adds is a structural observation — the solution set is not just a collection of vectors, it is a subspace. It contains the zero vector, it is closed under addition (if Ax = 0 and Ay = 0, then A(x + y) = 0), and it is closed under scalar multiplication. Every solution you find inherits this geometric structure.
The connection to subspaces from your prerequisite makes this concrete. When you row-reduce Ax = 0 and find free variables, each free variable corresponds to one independent direction within the null space. If there are two free variables, the null space is a plane through the origin; if one, it is a line; if none, only the zero vector satisfies Ax = 0. The number of free variables is the nullity — the dimension of nul(A). This is why nullity counts free variables precisely: each free variable parameterizes one independent direction in the solution set.
The term kernel (from linear transformation theory) refers to the same object. When you think of A not as a grid of numbers but as a function T(x) = Ax that maps Rn → Rm, the kernel is everything that maps to zero — the set of inputs that A "collapses." A large null space means A collapses many directions; a trivial null space (only the zero vector) means A is injective. The nullity therefore measures how much information A destroys.
To find a basis for nul(A), row-reduce to RREF, express pivot variables in terms of free variables, then write x as a linear combination of vectors — one per free variable. These vectors are the basis for nul(A). This procedure is mechanical, but the key insight is conceptual: the free variables are the degrees of freedom in the solution, and each one contributes exactly one dimension to the null space.