Linear Independence Calculator

Background.

Two arrows drawn from the same point in a plane either lie along one line or they do not. If they do, one is a stretched, shrunk or reversed copy of the other; the pair is linearly dependent, and adding multiples of them only ever reaches points on that line. If they do not, they are linearly independent, and combinations of the two reach every point in the plane. That is what makes a basis: two independent vectors are enough to describe any position in two dimensions.

The test is a single 2 × 2 determinant. Set the vectors side by side as the columns of a matrix and compute v1x·v2y − v1y·v2x. Exactly zero means dependent; any other value means independent. The determinant is also a measurement. Its absolute value is the area of the parallelogram the two vectors outline, and that area collapses to nothing precisely when they fall on one line. Its sign records orientation: positive when the second vector lies counter-clockwise from the first.

The same number decides whether a pair of linear equations has exactly one solution. The coefficients of x form one vector and the coefficients of y form the other; a zero determinant means the two lines are parallel or the same line, and Cramer's rule, which divides by the determinant, has nothing to work with.

The check is exact and limited to two vectors in two dimensions. Decimals are carried exactly as typed, so (0.1, 0.3) and (0.3, 0.9) are correctly found dependent, where ordinary binary arithmetic can leave a stray 1.39 × 10⁻¹⁷. The flip side is that there is no tolerance: a determinant of 0.000002 counts as independent even if the data behind it were never that precise.

What is linear independence calculator?

Vectors are linearly independent when none of them can be written as a combination of the others. For two vectors this reduces to a simple condition: neither is a scalar multiple of the other, and neither is the zero vector. In the plane, two independent vectors form a basis, meaning every other vector can be written as a·v1 + b·v2 in exactly one way. This page decides independence for one pair of real two-dimensional vectors from the determinant of the matrix whose columns they form.

How to use this calculator.

1. 01Enter the x and y components of the first vector, then those of the second. Each component can be a whole number, decimal or negative value up to one million in size.
2. 02To test a system of two linear equations, enter the coefficients of x from both equations as the first vector and the coefficients of y as the second.
3. 03Read the verdict and the determinant beside it. A nonzero determinant also gives the area of the parallelogram the vectors span, with its sign showing their orientation.
4. 04If the vectors come from measured or rounded data, do not stop at the verdict. Divide the determinant by the product of the two vector lengths, as described in the questions below, to see how close to parallel they really are.

The formula.

det = v1x × v2y − v1y × v2x. The vectors are the columns of the matrix [[v1x, v2x], [v1y, v2y]], and a 2 × 2 determinant is the product of the main diagonal minus the product of the other diagonal. If v2 is k times v1, then v2x = k·v1x and v2y = k·v1y, both products equal k·v1x·v1y, and their difference is zero; that is why zero means dependent. Geometrically, |det| = |v1| × |v2| × sin φ, where φ is the angle between the vectors, so the determinant grows with both their lengths and their spread. The verdict reads the exact value, but the determinant is displayed to ten decimal places, so one smaller than 0.00000000005 shows as 0 while the verdict still says independent.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.