Last updated October 4, 2026 · Report an error
A set of vectors is linearly independent when none of them can be written as a combination of the others — equivalently, when the matrix they form has full rank. This tool row-reduces that matrix and compares its rank to the number of vectors.
How it works
The vectors are stacked as rows of a matrix, which is then reduced to row echelon form via Gauss-Jordan elimination (the same verified helper used by the Matrix Rank calculator). If the resulting rank equals the number of vectors, they're independent; if it's smaller, they're dependent.
- Click Calculate to see your results.
Examples
An independent set
(1,0) and (0,1) form a matrix of rank 2, matching the 2 vectors — they're linearly independent.
A dependent set
(1,2) and (2,4) form a matrix of rank 1 (the second row is just double the first) — they're linearly dependent.
Who should use it
- Linear algebra coursework on linear independence, span, and basis.
- Checking whether a proposed basis for a vector space is actually valid.
Industry applications
- Linear algebra education
- Computer graphics and engineering (basis and coordinate systems)
Advantages
- Reuses the same verified row-reduction engine as the Matrix Rank calculator.
- Shows the full row-reduction trace, not just a yes/no answer.
Limitations
- Limited to 2-4 vectors in 2-4 dimensions for a fast, reliable computation.
Common mistakes to avoid
- Assuming vectors are independent just because none is a simple multiple of another — dependence can involve combinations of 3 or more vectors, not just pairs.
Best practices
- If you already suspect dependence, check whether one vector's components are proportional to another's first — it's the simplest case to spot by eye.
Tips
- A basis for an n-dimensional space needs exactly n linearly independent vectors — this tool is a direct way to verify that condition.