Column Space Calculator

The column space of a matrix A is the set of all linear combinations of its column vectors. It is a subspace of ℝᵐ (where m is the number of rows), representing every vector that can be produced by multiplying A by some vector. The dimension of the column space equals the rank of the matrix.

To find the column space, perform row reduction (Gaussian elimination) on the matrix to identify the pivot columns. The original columns of the matrix corresponding to those pivot positions form a basis for the column space. The number of pivot columns equals the rank.

A basis for the column space consists of the linearly independent columns from the original matrix that correspond to the pivot columns found after row reduction. These vectors span the column space and none of them can be expressed as a combination of the others.

The column space is spanned by the column vectors of a matrix and lives in ℝᵐ, while the row space is spanned by the row vectors and lives in ℝⁿ. Both have the same dimension (the rank of the matrix), but they are generally different subspaces.

The rank of a matrix equals the dimension of its column space — that is, the number of linearly independent columns. A higher rank means the columns span a larger subspace. For an m×n matrix, rank ≤ min(m, n).

Nullity is the dimension of the null space (kernel) of the matrix — the number of free variables after row reduction. By the Rank-Nullity Theorem, rank + nullity = n (the number of columns). So if you know the rank from the column space, nullity follows directly.

Yes — if the rank of the matrix equals m (the number of rows), then the column space spans all of ℝᵐ. This means the matrix has full row rank and every vector in ℝᵐ can be expressed as a linear combination of the columns.

Column space appears in linear regression (determining if a solution exists), computer graphics (transformations), machine learning (dimensionality reduction), and engineering (solving systems of linear equations). It tells you which output vectors are reachable by the linear transformation defined by the matrix.