Test whether a set of vectors is linearly independent. Enter 2–4 vectors in 2D or 3D, compute determinant, rank, row reduction (RREF), and find dependency relations if dependent.
Linear independence is one of the most fundamental concepts in linear algebra. 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 the trivial solution where all coefficients are zero.
This calculator tests linear independence for 2–4 vectors in 2D or 3D space. It performs Gaussian elimination to compute the row echelon form (RREF) of the matrix formed by the vectors, determines the rank, and calculates the determinant when the matrix is square. If the vectors are dependent, the tool finds and displays an explicit dependency relation showing which linear combination equals zero.
Understanding linear independence is essential across mathematics, physics, computer science, and engineering. In solving systems of linear equations, independent column vectors guarantee a unique solution. In machine learning, feature independence affects model quality. In physics, independent force vectors define the degrees of freedom of a system.
The visual component bars let you compare vector magnitudes and individual components side-by-side, making it easy to spot proportional vectors (a telltale sign of dependence). The RREF table shows the step-by-step reduction that reveals the rank and pivot structure of the matrix.
Linear Independence Calculator helps you solve linear independence problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter Vector 1, Vector 2, Vector 3 once and immediately inspect Linearly Independent?, Rank, Determinant to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Vectors {v₁, v₂, …, vₙ} are linearly independent iff rank([v₁|v₂|…|vₙ]) = n. For square matrices: independent iff det ≠ 0. Dependency relation: c₁v₁ + c₂v₂ + … + cₙvₙ = 0 with at least one cᵢ ≠ 0.
Result: Linearly Independent? shown by the calculator
Using the preset "Standard 2D basis", the calculator evaluates the linear independence setup, applies the selected algebra rules, and reports Linearly Independent? with supporting checks so you can verify each transformation.
This calculator takes Vector 1, Vector 2, Vector 3, Vector 4 and applies the relevant linear independence relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Linearly Independent?, Rank, Determinant, Dimension to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Vectors are linearly independent if none of them can be expressed as a linear combination of the others. The only way to combine them to get the zero vector is with all coefficients equal to zero.
For a square matrix (n vectors in n dimensions), a non-zero determinant means the vectors are linearly independent. A zero determinant means they are dependent. The determinant only applies when the number of vectors equals the dimension.
The rank is the number of linearly independent rows (or equivalently columns). It equals the number of pivot positions in row echelon form. Full rank means all rows/columns are independent.
No. In ℝ³, at most 3 vectors can be linearly independent. Any set of 4 or more vectors in 3D must be dependent — this follows from the dimension theorem.
A dependency relation is specific coefficients c₁, c₂, …, cₙ (not all zero) such that c₁v₁ + c₂v₂ + … + cₙvₙ = 0. It explicitly shows how the vectors are related.
Linear independence determines whether a system of equations has a unique solution, whether a set of vectors forms a basis, the dimension of a vector space, and whether transformations are invertible. It is foundational in linear algebra, differential equations, and data science.