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Linear Independence Calculator

Check if up to 5 vectors in ℝ², ℝ³, or ℝ⁴ are linearly independent. Finds the span dimension and a basis via RREF with full step-by-step Gaussian elimination.

Quick Answer
Vectors are linearly independent if rank = number of vectors (no vector is a combination of others). Use RREF: count non-zero rows. The independent ones form a basis for their span.
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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.

The RREF Test

The fastest computational test: form a matrix with the vectors as rows, then row-reduce to RREF.

  • Count the non-zero rows → this is the rank
  • If rank = number of vectors → linearly independent
  • If rank < number of vectors → linearly dependent (rank tells you how many are actually independent)

Span and Basis

The span of vectors is the set of all their linear combinations — always a subspace. A basis is a minimal set of independent vectors that span the same subspace. The dimension of the span equals the rank.

International Curriculum Notes

Linear independence is covered in:

  • MIT 18.06 (Gilbert Strang): Core of the entire course — columns of A, pivot columns, free variables
  • UK A-Level Further Maths (Edexcel/AQA): Introduced in the Matrices and Linear Transformations modules
  • IB HL Mathematics: Part of the linear algebra option; students test independence via determinant (for square cases) or RREF
  • Germany (Lineare Algebra): First-year university course; independence via the Wronskian for function spaces; vector spaces over arbitrary fields
  • Japan (線形代数): Covered in university — 基底 (basis), 次元 (dimension), 線形独立 (linear independence)

Key Theorem: Dimension Bound

In ℝⁿ, any set of more than n vectors must be linearly dependent. You cannot pack more than n independent vectors into n-dimensional space. This follows directly from the rank-nullity theorem: rank ≤ n, so at most n vectors can be independent.

Applications

  • PCA: Principal components are orthogonal (hence independent) directions of maximum variance
  • Solutions to linear systems: The null space basis vectors are linearly independent; they describe all solutions to Ax = 0
  • Signal processing: Independent basis functions (Fourier, wavelets) allow unique signal decomposition
  • Economics: Independent constraint vectors mean a unique optimum exists in linear programming

Frequently Asked Questions

What does linear independence mean?

c₁v₁ + … + cₙvₙ = 0 only when all cᵢ = 0. No vector is a linear combination of the others.

How do you check independence with RREF?

Write vectors as rows, row-reduce to RREF, count non-zero rows. rank = number of vectors → independent.

What is the span of vectors?

All linear combinations c₁v₁ + … + cₙvₙ. Always a subspace. Its dimension equals the rank.

What is the difference between span and basis?

Span is the full set of combinations. A basis is a minimal spanning set — independent vectors that generate the same space.

Can more vectors than the dimension be independent?

No. In ℝⁿ, at most n vectors can be linearly independent.

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