Find the column space basis and determine the rank of any matrix.
Last updated: 2026-05-24T22:58:31.717Z
Enter matrix entries as numbers; decimals allowed.
Rank
0
Nullity
3
Column Space Basis Vectors:
Zero matrix - no basis vectors
The column space (also called the range) of a matrix is the set of all linear combinations of its column vectors. It represents all possible outputs of the matrix when multiplied by any input vector. Geometrically for a 3×3 matrix, the column space can be a plane, a line, or just the origin.
A basis for the column space is a set of linearly independent vectors that spans all vectors in the column space. The rank of a matrix equals the dimension of its column space, which equals the number of pivot columns in row echelon form.
Find the column space basis of:
⎡ 1 2 3 ⎤
⎢ 2 4 6 ⎥
⎣ 3 6 9 ⎦
Step 1: Reduce to row echelon form
Step 2: Pivot columns are: 1 (column 1 only)
Step 3: Rank = 1, Nullity = 2
Column space basis: {[1, 2, 3]}
No. The zero vector is always in the column space, so it's never empty.
Nullity = Total columns - Rank. It's the dimension of the null space.
Row operations preserve column space, so pivot columns of the original matrix form the basis.
The column space spans the entire ℝⁿ (where n = number of rows). This matrix has full row rank.
Not always. For a 3×3 matrix, it could be 3D space, a plane, a line, or just a point (origin).
Ax = b has a solution only if b is in the column space of A.
No. By definition, basis vectors are linearly independent.
Row space uses row vectors; column space uses column vectors. They have equal dimension (rank).
Related Tools
Calculate adjoint matrix.
Find angle between vectors.
Find characteristic polynomial.
Cholesky factorization.
Expand determinant by cofactors.
Calculate cofactor matrix.