Linear Independence Calculator

A collection of vectors is linearly independent when the only coefficients that make their linear combination equal the zero vector are all zero. For three vectors v1, v2, and v3, independence means that c1v1 + c2v2 + c3v3 = 0 forces c1 = c2 = c3 = 0. If any nonzero choice of coefficients also produces the zero vector, the collection is linearly dependent. In practical terms, a dependent vector adds no new direction because it can be written as a combination of the others. This calculator places the three input vectors into a matrix and uses Gaussian elimination to find its rank. Elementary row operations expose the number of pivot positions without changing the linear relationships among the vectors. When the rank equals the number of vectors, every vector contributes a new pivot and the set is independent. When the rank is smaller than three, at least one vector is redundant and the set is dependent. The vectors must have the same dimension because components must occupy consistent coordinate positions. Dimension imposes an immediate limit. Three vectors in two-dimensional space must be dependent because a two-dimensional matrix can have rank at most two. Three vectors in three-dimensional or higher space may be independent, but they are not guaranteed to be. The familiar standard basis vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) are independent. By contrast, vectors (1, 2, 3) and (2, 4, 6) point along the same line, so any set containing both already includes a direct scalar relationship. Linear independence determines whether coordinates are unique, whether a matrix can be invertible, and whether equations contain redundant information. It is fundamental in solving systems, constructing bases, analyzing transformations, fitting models, and separating signals. To interpret the output, compare the reported rank with the number of input vectors. A full rank of three confirms independence; any smaller rank confirms dependence. Decimal inputs are supported, although values extremely close to zero are treated as numerical zero to keep floating-point roundoff from creating false pivots.