Four Fundamental Subspaces Calculator

Find bases for the column space1, nullspace2, row space3, and left nullspace4 of a matrix \(A\).
Enter matrix dimensions, generate the input matrix, fill in the values, and compute all four subspaces.

For an \(m \times n\) matrix \(A\):

Rank-Nullity Theorem:

  • \(\text{rank}(A) + \text{nullity}(A) = n\) (number of columns)
  • The dimension of the column space plus the dimension of the nullspace equals the number of columns.

Row Rank = Column Rank:

  • \(\text{rank}(A^T) = \text{rank}(A)\)
  • The dimension of the column space \(C(A)\) equals the dimension of the row space \(C(A^T)\).

Orthogonal Complements5:

  • \(C(A) \oplus N(A^T) = \mathbb{R}^m\) and \(C(A^T) \oplus N(A) = \mathbb{R}^n\)
  • The column space and left nullspace are orthogonal complements in \(\mathbb{R}^m\). The row space and nullspace are orthogonal complements in \(\mathbb{R}^n\).

This calculator uses RREF with partial pivoting6 to identify pivot and free columns, then extracts bases for all four fundamental subspaces.

Set matrix dimensions

Enter dimensions and click Generate Matrix to create the input grid.

Display values as:
1 — Set Up
Loads a pre-filled 3×4 matrix — ready to compute.
2 — Compute
Finds bases for the column space, nullspace, row space, and left nullspace via RREF.
Resets all dimensions, matrix entries, and results.
3 — Random
Fills the matrix with random integers and cycles through different ranks.
Export
Downloads the computed subspace bases as a ready-to-compile LaTeX document.