Linear Algebra Intermediate

Nullity & Rank-Nullity Theorem

Compute the nullity (dimension of the null space) of a matrix. Row reduce to RREF, identify free variables, find null space basis vectors, and verify rank + nullity = n.

Subject: Linear Algebra
Input: Up to 3×4 matrix
Output: nullity, null space basis, RREF
Live
Matrix Input
Result
Enter matrix values and click Compute Nullity

Step-by-Step Work

The Rank-Nullity Theorem
rank(A) + nullity(A) = n (cols)

The null space (kernel) of A is the set of all vectors x such that Ax = 0. Its dimension is the nullity. Each free variable in RREF corresponds to one basis vector of the null space.

To find a basis: for each free variable, set it to 1 and all other free variables to 0, then back-substitute for pivot variables.

Exam tip — if the null space is just {0}, the system Ax = 0 has only the trivial solution, so the columns of A are linearly independent.
Null Space Facts
  • Null space = kernel = solution set of Ax = 0
  • Each free variable → one basis vector
  • nullity = 0 ⟺ columns are linearly independent
  • For square A: nullity > 0 ⟺ A is singular
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