# Linear Independence Calculator

URL: https://speedycalculator.com/calculators/math/linear-independence/
Last updated: 2026-10-30

> The linear independence calculator checks whether up to four vectors (with up to four coordinates) are linearly independent and gives the dimension of the space they span.

## Inputs

- Number of vectors
- Number of coordinates
- v₁: x
- v₁: y
- v₁: z
- v₁: w
- v₂: x
- v₂: y
- v₂: z
- v₂: w
- v₃: x
- v₃: y
- v₃: z
- v₃: w
- v₄: x
- v₄: y
- v₄: z
- v₄: w

## Results

- Result
- Dimension of the span (rank)
- Row-reduced matrix
- Why

## Formula

```
Vectors are independent ⇔ rank of the matrix with them as rows = number of vectors
```

## Explanation

Independent vectors carry no redundant information – none is a combination of the others. Checking independence tells you whether vectors form a basis, whether data features are collinear and whether a system has a unique solution.

### How to use the Linear Independence Calculator

- **Number of vectors**
- **Number of coordinates**
- **v₁: x**
- **v₁: y**
- **v₁: z**
- **v₁: w**
- **v₂: x**
- **v₂: y**
- **v₂: z**
- **v₂: w**
- **v₃: x**
- **v₃: y**
- **v₃: z**
- **v₃: w**
- **v₄: x**
- **v₄: y**
- **v₄: z**
- **v₄: w**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The vectors are placed as rows of a matrix and reduced with Gauss–Jordan elimination. The number of non-zero rows (the rank) is the dimension of their span.

## Worked example

Inputs — Number of vectors: 3; Number of coordinates: 3; v₁: x: 1; v₁: y: 2; v₁: z: 3; v₂: x: 2; v₂: y: 4; v₂: z: 6; v₃: x: 0; v₃: y: 1; v₃: z: 1.

Results — **Result = Linearly dependent**; Dimension of the span (rank) = 2; Row-reduced matrix = ⎡ 1 0 1 ⎤
⎢ 0 1 1 ⎥
⎣ 0 0 0 ⎦; Why = The rank (2) is less than the number of vectors (3), so at least one vector is a combination of the others..

## Key terms

- **Linearly independent**: No vector is a linear combination of the others.

## FAQ

**Can three vectors in 2D be independent?**
No – at most two vectors in ℝ² can be independent.

**Is a set containing the zero vector independent?**
Never.

**How do I check two vectors?**
They are dependent only if one is a multiple of the other.

**What is the connection with determinants?**
n vectors in ℝⁿ are independent exactly when their determinant is non-zero.


## Sources

- [Wolfram MathWorld – Linearly independent](https://mathworld.wolfram.com/LinearlyIndependent.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/)