How to Determine Linear Independence
A set of vectors v₁, v₂, ..., vₖ is linearly independent if the only way to combine them with scalars c₁, c₂, ..., cₖ to get the zero vector is the trivial combination: c₁v₁ + c₂v₂ + ... + cₖvₖ = 0 forces c₁ = c₂ = ... = cₖ = 0. If any other combination (not all zero) also produces the zero vector, the set is linearly dependent — at least one vector can be written as a linear combination of the others. This calculator checks 2 or 3 vectors in ℝ² or ℝ³ using the general rank test and, when the system is square, the determinant shortcut.
How the calculation works
The calculator stacks your vectors as the rows of a matrix and reduces that matrix to row-echelon form using Gaussian elimination with partial pivoting (swapping in the largest available pivot at each step for numerical stability). The rank is the number of nonzero pivot rows left after elimination. If the rank equals the number of vectors k, every vector contributes a new, independent direction and the set is linearly independent; if the rank is less than k, some vectors are redundant and the set is dependent. When the number of vectors equals the dimension (a square system), the calculator also reports the determinant: for 2 vectors in ℝ², det = a·d − b·c; for 3 vectors in ℝ³, the calculator expands along the first row using the standard 3×3 cofactor formula. A nonzero determinant is equivalent to a full rank and confirms independence; a zero determinant confirms dependence.
Common mistakes
- Applying the determinant test to a non-square system: the determinant is only defined for a square matrix, so you can only use it directly when the number of vectors equals the dimension (e.g., 2 vectors in ℝ² or 3 vectors in ℝ³). For any other combination, use the rank test instead.
- Assuming more vectors always add information: in an n-dimensional space, any set of more than n vectors is automatically dependent, since the rank can never exceed n. Three vectors in ℝ² will always be dependent.
- Confusing "nonzero vectors" with "independent vectors": two nonzero vectors can still be dependent if one is a scalar multiple of the other (they point along the same line).
Real-world applications
- Solving linear systems: a square coefficient matrix has a unique solution exactly when its rows (or columns) are linearly independent, i.e. its determinant is nonzero.
- Basis and dimension: a set of n linearly independent vectors in ℝⁿ forms a basis — every other vector in the space can be written uniquely as a combination of them.
- Computer graphics and physics: independent direction vectors (e.g., for coordinate axes or force components) avoid redundant or degenerate transformations.
- Data science: checking whether feature vectors are independent helps detect multicollinearity before fitting a linear model.