Null Space & Nullity

null(A) = x : Ax = 0. Nullity = its dimension.

The idea

The null space of a matrix $A$ records which vectors the matrix sends to zero. Multiplication by $A$ can flatten whole directions of input onto the origin, and the inputs it flattens are the solutions of $Ax = 0$; a vector $w$ outside that set keeps a nonzero image $Aw$.

Those solutions form a subspace: if $Ax = 0$ and $Ay = 0$, then $A(x + y) = 0$ and $A(cx) = 0$ as well, so the solution set is closed under addition and scaling — a line, a plane, or a higher-dimensional flat through the origin. We define the null space of $A$ as $\text{null}(A) = \{x : Ax = 0\},$ and its dimension is the nullity of $A$.

To find the nullity, solve $Ax = 0$ and count the free parameters in the solution: each is an independent direction the matrix sends to zero, and each contributes one dimension.

Ways to work on it

• Walkthrough. Compute nullity for identity, rank-1, and zero matrices.
• Practice. Find nullity of a 2 × 3 matrix.
• Hardest. Find the nullity of a matrix using the rank-nullity theorem.

Builds on

• Matrix Rank

Leads to

• Eigenvectors & Eigenspaces
• Rank-Nullity Theorem

Not sure where to start? Take the ten-question placement test.