🔍 Linear Independence Calculator
Check if vectors are linearly independent
How to Use This Calculator
Select Number of Vectors
Choose how many vectors to test (2-5 vectors).
Choose Dimension
Select vector dimension (2D, 3D, 4D, or 5D).
Enter Vectors
Input the components of each vector.
Get Result
See if vectors are linearly independent or dependent.
Formula
Vectors are LI ⟺ Rank = Number of Vectors
Checked via Gaussian elimination to RREF
Linear Independence:
Vectors v₁, v₂, ..., vₙ are linearly independent if the only solution to c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is c₁ = c₂ = ... = cₙ = 0.
Linear Dependence:
Vectors are linearly dependent if at least one can be written as a combination of others: c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 with not all cᵢ = 0.
Test Method:
- Arrange vectors as columns in a matrix
- Perform Gaussian elimination to RREF
- Count pivot columns (rank)
- Rank = number of vectors → linearly independent
- Rank < number of vectors → linearly dependent
About Linear Independence Calculator
The Linear Independence Calculator determines whether a set of vectors is linearly independent or dependent. It uses Gaussian elimination to compute the rank, which reveals if vectors are independent.
When to Use This Calculator
- Linear Algebra: Test vector independence
- Basis Finding: Verify if vectors form a basis
- Span Analysis: Check if vectors span full dimension
- Vector Spaces: Understand vector relationships
- Dimensions: Find dimension of spanned space
Why Use Our Calculator?
- ✅ Gaussian Elimination: Uses RREF method
- ✅ Clear Result: Shows independent/dependent status
- ✅ Rank Display: Shows computed rank
- ✅ Educational: Helps understand independence
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Linear Independence: No vector is a combination of others
- Linear Dependence: At least one vector is redundant
- Rank: Number of linearly independent columns
- Basis: Maximal linearly independent set
- Dimension: Number of vectors in a basis
Applications
Basis Finding: Find a basis for the span of given vectors.
System Solving: Determine if system of linear equations has unique or infinite solutions.
Frequently Asked Questions
What does linearly independent mean?
Vectors are linearly independent if no vector can be written as a linear combination of the others. No redundant information exists.
Can I have more vectors than dimension?
Yes, but if you have more vectors than dimension, they are always linearly dependent. For n-dimensional space, at most n vectors can be independent.
Are the zero vector and any other vector independent?
No! Any set containing the zero vector is linearly dependent because 0 can be written as 0·v₁ + 0·v₂ + ... + 1·0 = 0.
What is the difference between independence and orthogonality?
Independence means no linear dependence. Orthogonality means perpendicular (dot product = 0). Orthogonal vectors are always independent, but independent vectors need not be orthogonal.
If rank equals number of vectors, are they independent?
Yes! Rank = number of vectors is the precise condition for linear independence. Each vector contributes to the rank (no redundancy).