The column space is the set of all vectors Ax – everything the matrix can “reach”. Its dimension is the rank. It tells you whether Ax = b has a solution: exactly when b is in the column space.
How to use the Column Space Calculator
- Choose the matrix size – the matching grid of fields appears.
- Enter the entries row by row (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The matrix is reduced to reduced row echelon form. The pivot columns of the original matrix form a basis of the column space.
Formula
Col(A) = span of the columns of A; basis = columns with pivots in rref(A)
Worked example
Inputs — Matrix size (rows × columns): 3 × 3; a₁: 1; b₁: 2; c₁: 3; a₂: 2; b₂: 4; c₂: 6; a₃: 1; b₃: 0; c₃: 1.
Results — Basis of the column space = (1, 2, 1) (column 1)
(2, 4, 0) (column 2); Dimension (rank) = 2; Reduced row echelon form = ⎡ 1 0 1 ⎤
⎢ 0 1 1 ⎥
⎣ 0 0 0 ⎦.
Key terms
- Rank
- The dimension of the column space.
Frequently asked questions
What is a column space?
The span of a matrix's columns, a subspace of ℝᵐ.
Why use the original columns, not the reduced ones?
Row operations change the column space; only the pivot positions carry over.
How is it related to the null space?
rank + nullity = number of columns.
When is the column space all of ℝᵐ?
When the rank equals the number of rows.