Finding Null Spaces

Linear Algebra (Matrices) · Axiom Academy

Solve by row reduction, parametrize the free variables, and write the null space as a span of basis vectors. Find the null space of the matrix A — that is, every vector satisfying — and express it as . We expect a flat subspace through the origin. Two free variables will give two basis vectors, so is a 2-dimensional plane. Nicely done — you found the null space of A from start to finish. The moves that got us there: The null space solves : it is the set of all vectors A sends to , and it is always a subspace of . Row reduction does the work: reduce to RREF; the zero right-hand column never changes, so you can reduce A alone. Pivots vs. free columns: columns with a leading 1 are pivot variables ( x_1, x_3 ); columns without one are free ( x_2, x_4 ). Parametrize the free variables: set , solve the pivots, and split into one vector per parameter. Basis & dimension: one basis vector per free variable, so here (this is the nullity, and rank + nullity =4 ). Always verify: and confirm the basis is correct. The null space is everywhere in linear algebra — homogeneous systems, kernels of linear maps, and eigenspaces (the null space of ) are all the same computation.

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