Check if vectors are linearly dependent or independent
Enter 2 Vectors in 2D (each row = vector)
Enter 3 Vectors in 3D (each row = vector)
Result
Step-by-Step Derivation
Linear Dependence Definition
Dependent: c1v1 + c2v2 + ... = 0 (not all ci=0)
Independent: only trivial solution (all ci=0)
For n vectors in R^n: det≠0 → independent
Rank < n → dependent
A set of vectors is linearly dependent if at least one vector can be written as a linear combination of the others. Linearly independent vectors cannot be expressed in terms of each other and form the smallest spanning set (a basis) for their subspace.
⚠Any set containing more than n vectors in n-dimensional space is always linearly dependent. This is a fundamental result in linear algebra.
What is Linear Dependence?
Linear dependence and independence are fundamental concepts in linear algebra that describe relationships between vectors in a vector space. A set of vectors is linearly dependent if there exists a non-trivial linear combination that equals the zero vector. Otherwise, they are linearly independent.
Trivial Solution
The solution where all coefficients are zero. Always exists, but doesn't indicate dependence.
Determinant Test
For n×n matrix: det≠0 → independent; det=0 → dependent. Works for square matrices only.
Rank Test
Row echelon form: rank < number of vectors → dependent. Works for any dimensions.
Dimension Count
n vectors in m-dimensional space: n > m → always dependent. n = m: use determinant.
Teaching Example: v1=[1,2], v2=[3,4].
1. Matrix = [[1,3],[2,4]] (vectors as columns)
2. det = 1*4 - 3*2 = 4 - 6 = -2 ≠ 0
3. Conclusion: Linearly independent ✓
4. Basis for R²: can express any vector as a1v1 + a2v2.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.