Linear Independence Calculator
Check if vectors are linearly independent using matrix rank and determinant calculations
Vector Input
Vector 1
Vector 2
Linear Independence Analysis
Enter coordinates for the vectors to check linear independence
Example: v₁ = (1, 2, 3), v₂ = (4, 5, 6), v₃ = (7, 8, 9)
Example: 2D Linear Independence
Input Vectors
v₂ = (0, 1)
Analysis
Step 1: Form matrix A = [1 0; 0 1]
Step 2: Calculate determinant: det(A) = 1×1 - 0×0 = 1
Step 3: Since det(A) ≠ 0, vectors are linearly independent
Step 4: Matrix rank = 2, equal to number of vectors
Result: Linearly independent, span ℝ²
Linear Independence Concepts
Definition
Vectors are independent if no vector can be written as a combination of others
Only trivial solution to c₁v₁ + c₂v₂ + ... = 0
Matrix Rank
Rank equals maximum number of independent vectors
Found using Gaussian elimination
Determinant Test
For square matrices: det ≠ 0 means independent
Only works when #vectors = dimension
Quick Reference
Independence Test
rank(A) = number of vectors
Square Matrix
Independent ⟺ det(A) ≠ 0
Span Dimension
dim(span) = rank(A)
Maximum Vectors
At most n independent vectors in ℝⁿ
Understanding Linear Independence
What is Linear Independence?
Vectors are linearly independent if none of them can be expressed as a linear combination of the others. This means the only solution to c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 is when all coefficients cᵢ = 0 (the trivial solution).
Why It Matters
- •Basis formation: Independent vectors can form a basis for their span
- •Efficiency: No redundant vectors in the set
- •Unique representation: Each vector in the span has unique coefficients
- •System solvability: Related to existence of unique solutions
Applications
- •Computer graphics and 3D modeling
- •Data analysis and principal component analysis
- •Physics and engineering simulations
- •Machine learning feature selection
- •Economic modeling and optimization
Key Insight: In an n-dimensional space, you can have at most n linearly independent vectors. Any set of more than n vectors in ℝⁿ must be linearly dependent.
Testing Methods
Matrix Rank Method:
- • Form matrix with vectors as columns
- • Use Gaussian elimination to find rank
- • Independent if rank = number of vectors
- • Works for any number of vectors
Determinant Method:
- • Only for square matrices (same # vectors & dimensions)
- • Calculate determinant of matrix
- • Independent if determinant ≠ 0
- • Quick test but limited applicability