# Column Space Calculator

URL: https://speedycalculator.com/calculators/math/column-space/
Last updated: 2026-10-30

> The column space calculator finds a basis for the column space (range) of a matrix of up to 4 × 4 and its dimension, the rank, using Gauss–Jordan elimination.

## Inputs

- Matrix size (rows × columns)
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄

## Results

- Basis of the column space
- Dimension (rank)
- Reduced row echelon form

## Formula

```
Col(A) = span of the columns of A;  basis = columns with pivots in rref(A)
```

## Explanation

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.

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Column space](https://mathworld.wolfram.com/ColumnSpace.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)