About Linearly Dependant Vectors
Linear dependence explains whether one vector can be made from others. A set is dependant when at least one nonzero coefficient combination gives the zero vector. This calculator treats each matrix column as one vector. Rows represent vector components. It then reduces the matrix to row echelon form and measures rank.
Why This Test Matters
Dependence appears in algebra, geometry, statistics, physics, graphics, and engineering models. Independent vectors carry separate information. Dependant vectors repeat information already contained in the set. This distinction helps you detect redundant equations, weak coordinate systems, and unstable model inputs. It also supports basis selection, span checks, and dimension work.
What The Calculator Checks
The tool reads any rectangular matrix within the selected size. It finds pivot columns through Gauss-Jordan elimination. The number of pivots is the rank. If rank is smaller than the number of vectors, the vectors are linearly dependant. If every vector column is a pivot column, the vectors are independent. For square matrices, the determinant also gives a useful clue. A zero determinant means the square column set is dependant.
Understanding The Relation
A dependant set has a nontrivial relation. That means coefficients exist that are not all zero. When multiplied by the matching vectors and added, they produce the zero vector. The calculator builds nullspace basis vectors from the reduced matrix. Each basis vector is a possible coefficient pattern. This makes the answer more useful than a simple yes or no.
Practical Tips
Enter exact integers when possible. Use decimals only when your data requires them. A small tolerance helps handle rounding error. Larger tolerance values may treat tiny pivots as zero. That is useful for measured data, but it can change the decision. Review the reduced matrix and pivot list before using the result in formal work.
Best Use Cases
Use this page for homework checks, system analysis, coordinate validation, feature screening, and quick teaching examples. It is also helpful when you need a readable report. Export the result as a spreadsheet file or portable document. Keep the input matrix beside the reduced form. That makes each conclusion easier to verify.
For best results, compare several examples and note how pivots change with each new vector.