---
title: "Linear Independence Calculator — Span, Basis & Rank"
description: "Check if up to 5 vectors (ℝ², ℝ³, ℝ⁴) are linearly independent using RREF. Finds the span, basis, and dimension. Shows rank-nullity reasoning and step-by-step Gaussian elimination."
url: "https://worldcalculators.org/calculators/linear-independence/"
canonical: "https://worldcalculators.org/calculators/linear-independence/"
---

# Linear Independence Calculator

Check if up to 5 vectors (ℝ², ℝ³, ℝ⁴) are linearly independent using RREF. Finds the span, basis, and dimension. Shows rank-nullity reasoning and step-by-step Gaussian elimination.

## Quick answer

Vectors v₁, v₂, …, vₙ are linearly independent if the equation c₁v₁ + c₂v₂ + … + cₙvₙ = 0 has only the trivial solution c₁ = c₂ = … = cₙ = 0. In plain terms: no vector in the set can be written as a combination of the others.

## What Is Linear Independence?

Vectors v₁, v₂, …, vₙ are linearly independent if the equation c₁v₁ + c₂v₂ + … + cₙvₙ = 0 has only the trivial solution c₁ = c₂ = … = cₙ = 0. In plain terms: no vector in the set can be written as a combination of the others.
If a non-trivial solution exists (some cᵢ ≠ 0), the vectors are linearly dependent — at least one is redundant.

## Frequently Asked Questions

c₁v₁ + … + cₙvₙ = 0 only when all cᵢ = 0. No vector is a linear combination of the others.
Write vectors as rows, row-reduce to RREF, count non-zero rows. rank = number of vectors → independent.

Source: https://worldcalculators.org/calculators/linear-independence/
Markdown mirror of the HTML page. Prefer this URL for RAG; the interactive calculator still lives on the HTML page.