Linear Independence Calculator – Check Vector Independence
Check if vectors are linearly independent or dependent.
Table of Contents
How to Use
- Enter each vector on a separate line with components separated by commas or spaces
- For example: 1, 2, 3 on one line and 4, 5, 6 on the next
- Click calculate to determine if the vectors are linearly independent
- Review the rank, determinant (for square matrices), and RREF
What is Linear Independence?
A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. Equivalently, the only solution to c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is when all coefficients c₁, c₂, ..., cₙ are zero.
If at least one non-trivial combination exists (some coefficients are non-zero), the vectors are linearly dependent.
How to Check Linear Independence
- Form a matrix with the vectors as rows (or columns)
- Apply Gaussian elimination to reduce to row echelon form
- Count the number of non-zero rows (the rank)
- If rank equals the number of vectors, they are linearly independent
For square matrices, you can also check the determinant: if det ≠ 0, the vectors are independent.
Applications of Linear Independence
- Finding bases for vector spaces
- Solving systems of linear equations
- Determining if a transformation is invertible
- Signal processing and data compression
- Machine learning feature selection
Frequently Asked Questions
- What does it mean if vectors are linearly dependent?
- Linearly dependent vectors contain redundancy—at least one vector can be expressed as a combination of the others. This means they don't span as many dimensions as there are vectors.
- Can more vectors than the dimension be independent?
- No. In an n-dimensional space, at most n vectors can be linearly independent. Any set with more than n vectors must be dependent.
- What is the relationship between rank and independence?
- The rank of a matrix equals the maximum number of linearly independent rows (or columns). If you have k vectors and the rank is k, all vectors are independent.