Column Space Calculator — Matrix Rank & Basis
Find the column space of a matrix — its dimension (rank), basis vectors, and nullity — using Gaussian elimination. Enter any matrix up to 4×4, select a size, and fill in the entries.
Matrix size (rows × columns)
Matrix entries
3
Basis vectors for Col(A) (pivot columns)
How does this calculator work?
The column space of an m×n matrix has dimension equal to the rank (number of pivot columns after row reduction). Basis vectors are the pivot columns of the original matrix. Rank + Nullity = n (Rank–Nullity Theorem). Use row reduction to find pivots; this calculator does it automatically.
Formula
How this is calculated
The column space (or range) of an m×n matrix A is the set of all vectors that can be written as Ax for some x — equivalently, all linear combinations of A's column vectors. Its dimension is the rank of A, found by counting the pivot columns in the row-reduced form.
The calculator performs Gaussian elimination with partial pivoting: it finds the leading 1 in each column (the pivot), swaps rows to bring the largest-magnitude entry to the pivot position for numerical stability, then eliminates above and below. Columns containing a pivot are called pivot columns, and the corresponding columns of the original matrix (not the reduced form) form a basis for the column space.
The Rank–Nullity Theorem guarantees that rank(A) + nullity(A) = n, where n is the number of columns. The null space (kernel) has dimension nullity — the number of free variables after reduction. This calculator shows rank, nullity, and the basis vectors but does not enumerate all elements of the column space (which is infinite unless rank = 0). Note that floating-point entries may produce small numerical errors; entries very close to zero are treated as zero.
Frequently asked questions
The column space consists of linear combinations of the column vectors (lives in ℝ^m); the row space consists of linear combinations of the row vectors (lives in ℝ^n). Both have the same dimension (the rank), but they are generally different subspaces in different ambient spaces.
Row-reduce the matrix to echelon form, identify the pivot columns (those with a leading entry), then take those same columns from the ORIGINAL matrix — not the reduced one. The reduced columns are orthonormal in an algebraically convenient sense, but the original pivot columns span the same space with the geometric meaning intact.
Nullity is the dimension of the null space — the number of linearly independent vectors x satisfying Ax = 0. By the Rank–Nullity Theorem, rank + nullity always equals the number of columns. A matrix with nullity 0 has only the trivial solution and the columns are linearly independent.
Also known as
TG we-Calculate Editorial Team. (2026). Column Space Calculator — Matrix Rank & Basis [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/column-space-calculator
TG we-Calculate Editorial Team. "Column Space Calculator — Matrix Rank & Basis." TG we-Calculate. 2026. https://we-calculate.com/calculator/column-space-calculator.
TG we-Calculate Editorial Team, "Column Space Calculator — Matrix Rank & Basis," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/column-space-calculator
@misc{wecalculate_column_space_calculator, title = {Column Space Calculator — Matrix Rank & Basis}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/column-space-calculator}}, year = {2026}, note = {TG we-Calculate} }
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