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.
Formula
Vectors are independent ⇔ rank of the matrix with them as rows = number of vectors
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.
Frequently asked questions
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.