Null Space & Column Space Calculator
Find the null space (kernel) and column space basis of any matrix. Shows rank, nullity, all pivot columns, and the basis vectors extracted from RREF.
Matrix A (3×4)
| Subspace | Also called | Dimension | Found from |
|---|---|---|---|
| Null space | Kernel, Nul(A) | Nullity = n − rank | Free columns of RREF |
| Column space | Range, Im(A) | Rank | Pivot columns of ORIGINAL A |
| Row space | Row span | Rank | Non-zero rows of RREF |
| Left null space | Nul(Aᵀ) | m − rank | Null space of Aᵀ |
The Four Fundamental Subspaces
Gilbert Strang's "four fundamental subspaces" framework shows how an m×n matrix A creates four orthogonally complementary subspaces that together describe everything about the linear map A represents:
- Column space C(A) ⊂ ℝᵐ — dim = rank(A)
- Left null space N(Aᵀ) ⊂ ℝᵐ — dim = m − rank(A) · orthogonal complement of C(A)
- Row space C(Aᵀ) ⊂ ℝⁿ — dim = rank(A)
- Null space N(A) ⊂ ℝⁿ — dim = n − rank(A) · orthogonal complement of C(Aᵀ)
Frequently Asked Questions
When does Ax = b have a solution?
Ax = b has a solution iff b lies in the column space of A. If b is not in C(A), the system is inconsistent (no solution). If b is in C(A) and the null space is trivial 0, the solution is unique. If the null space has positive dimension (nullity > 0), there are infinitely many solutions: x = x_particular + any vector in N(A).
What is the difference between null space and kernel?
"Null space" and "kernel" are the same thing — different terminology from different traditions. US linear algebra texts (Strang, Anton) typically say "null space." European functional analysis texts say "kernel" (German: Kern, French: noyau). Both mean the set of all x satisfying Ax = 0. This calculator uses both terms interchangeably.