The linear independence calculator above takes a set of vectors and answers the defining question: is there a combination of them, with coefficients not all zero, that adds up to the zero vector? If there is, the set is dependent and the tool prints the actual coefficients. If there is not, the set is independent and every vector in it contributes a direction the others cannot reach. Both answers come from Gauss-Jordan elimination on the matrix whose columns are the vectors, with a tolerance you can set to suit exact or measured data.
Arb Digital publishes this because the yes-or-no answer is rarely the useful part. Knowing a set is dependent tells you nothing about which vector is redundant or by how much; the dependency relation tells you exactly that. In practice — regression design matrices, feature sets, force diagrams, colour spaces — the question that matters is which column is a combination of the others, and that is what the relation identifies.
What This Linear Independence Calculator Does
It builds a matrix with your vectors as columns, reduces it to row echelon form with partial pivoting, and counts the pivots. That count is the rank, and it equals the dimension of the span. The set is independent precisely when the rank equals the number of vectors. When it is smaller, the tool extracts a vector from the null space and reports it as a dependency relation with explicit coefficients, so you can see which vectors combine and in what proportions.
Three neighbouring tools do adjacent jobs. Our determinant calculator gives a yes-or-no independence test, but only for the special case of exactly n vectors in n dimensions where the matrix is square — a non-zero determinant means independent. Our matrix rank calculator computes the rank of any matrix, which is the same underlying quantity but framed as a property of the matrix rather than of a set of vectors. Our null space calculator finds the complete set of solutions to Ax = 0, of which the single relation shown here is one member. This page is the vector-set framing with the dependency relation as the headline output.
How to Use It
- Enter one vector per line. Components separated by commas or spaces. Two, three, four or more dimensions all work.
- Keep the dimensions consistent. Every vector must have the same number of components, and the tool will tell you which line disagrees if one does not.
- Set a tolerance that matches your data. Exact integers can use the tightest setting; anything measured or rounded needs a looser one, for the reason described below.
- Read the relation, not just the verdict. If the set is dependent, the coefficients tell you exactly which vector can be dropped without shrinking the span.
- Check the nullity. It counts how many independent relations exist, which is how many vectors are surplus to a basis.
The Method and How It's Calculated
A set of vectors v₁ … vk is linearly independent when the only solution to c₁v₁ + c₂v₂ + … + ckvk = 0 is all coefficients zero. Written as a matrix equation that is Ac = 0, where A has the vectors as its columns. So independence is exactly the statement that A has a trivial null space, and by the rank-nullity theorem that means rank(A) = k.
The computation is Gauss-Jordan elimination with partial pivoting. Each step selects the largest available entry in the current column as the pivot, normalises its row, and eliminates that column from every other row. Columns that end up containing a pivot correspond to vectors that are independent of the ones before them; columns without a pivot are the free ones, and each free column produces a dependency relation. To build it, set the free coefficient to 1 and read the negated entries in the pivot rows as the remaining coefficients.
The default set makes this concrete. With columns (1, 2, 3), (4, 5, 6) and (7, 8, 9), elimination gives pivots in the first two columns and reduces the third to (−1, 2, 0). Column three is free, so set c₃ = 1, which forces c₁ = 1 and c₂ = −2. The relation is v₁ − 2v₂ + v₃ = 0, and substituting confirms it: (1 − 8 + 7, 2 − 10 + 8, 3 − 12 + 9) = (0, 0, 0). Rank 2, nullity 1, span a plane rather than all of three-dimensional space. MIT OpenCourseWare's Independence, Basis and Dimension materials work through the same elimination argument in detail.
The Counting Rule That Settles Most Cases
Before running any elimination, one comparison resolves a large share of real questions: if you have more vectors than components, the set is dependent. Always. Four vectors in three-dimensional space, ten features derived from eight measurements, six colours mixed from three primaries — all dependent by counting alone, because the rank cannot exceed the number of rows and therefore cannot reach the number of columns.
The converse does not hold. Having no more vectors than components does not make a set independent; it only makes independence possible. Three vectors in three-dimensional space can still be dependent, as the default example shows. And a set containing the zero vector is always dependent, because the zero vector can be given any non-zero coefficient while everything else takes zero. Two vectors are dependent precisely when one is a scalar multiple of the other, which is the only case simple enough to check by eye.
Why Tolerance Is Not a Detail
Exact linear dependence is a knife edge. Perturb any coefficient in a dependent set by the smallest amount and it becomes independent — mathematically, unambiguously independent, with a full-rank matrix and a trivial null space. That is why a tolerance exists at all: without one, floating-point arithmetic would report almost every real data set as independent, including sets that are dependent in every way that matters.
The default set illustrates it. Change the last component of the third vector from 9 to 9.000000001 and the exact answer flips to independent. At the tightest tolerance the tool will report that. At the loosest it correctly reports dependence, because a relation that holds to one part in a billion is a genuine relation in any applied setting. The right threshold depends on where your numbers came from: exact integers from a specification can use the tightest setting, values rounded to four decimal places need something near 1e−4, and sensor readings need a threshold set by their own precision. Reporting "independent" on data that is dependent to within measurement error is a false answer dressed as a rigorous one.
Multicollinearity: the Same Problem Under Another Name
In regression, near-dependence among predictor columns is called multicollinearity, and it is the most common practical encounter with this calculation. Exact dependence — including a variable and its duplicate, or all categories of a dummy-coded variable plus an intercept — makes the normal equations unsolvable outright. Near-dependence is worse in a way, because the model still fits and reports coefficients, but those coefficients become wildly unstable: small changes in the data swing them enormously, and their signs can flip.
The mechanism is the near-zero pivot. When a column is almost a combination of the others, the elimination divides by something very small, and everything downstream inherits an enormous sensitivity. Running your design matrix through this tool with a loose tolerance is a quick diagnostic — if it reports dependence, the relation names precisely which predictors are duplicating each other. Our linear regression calculator fits the model itself, and MIT's 18.06 Linear Algebra course covers the four fundamental subspaces that explain why the instability arises.
Rank, Span and Basis: Reading the Grid
The four numbers in the grid are tightly linked, and understanding the links makes the output much more useful. The rank is the number of pivots. The dimension of the span equals the rank — a dependent set of five vectors with rank 3 spans exactly the same space as some three of them would. The nullity is the number of vectors minus the rank, and it counts how many independent dependency relations exist, which is how many vectors are surplus.
A basis for the span is any maximal independent subset, and the pivot columns give you one directly: keep the vectors in pivot positions, discard the free ones, and you have a basis with no loss of span. Note that the choice is not unique — different pivoting orders give different valid bases — but the size is always the same, and that invariance is what makes dimension a well-defined property rather than an artefact of how you did the arithmetic. Our vector calculator handles the individual operations, and our matrix calculator covers products and transposes on the assembled matrix.
Arb Digital's analysts check for collinear inputs before fitting anything, so the coefficients you act on are stable ones.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Assuming equal counts mean independence — three vectors in three dimensions can still be dependent, and only elimination will tell you.
- Forgetting the zero vector — any set containing it is dependent, because that vector can take a non-zero coefficient while all the others take zero.
- Using an exact tolerance on measured data — floating-point noise will report near-dependent columns as independent, which is technically true and practically useless.
- Mixing vector dimensions — every vector needs the same number of components, or the matrix is not well formed and the result is meaningless.
- Treating the dependency relation as unique — when the nullity is above one there are many valid relations, and this tool shows one of them rather than all.
Related Free Tools From Arb Digital
Test the square case with the determinant calculator, get the rank of any matrix with the matrix rank calculator, find every solution of Ax = 0 with the null space calculator, work with individual vectors using the vector calculator, or multiply and transpose with the matrix calculator. Everything we publish is listed on the free online tools hub.
Frequently Asked Questions
A set of vectors is linearly independent when the only way to combine them into the zero vector is to give every coefficient the value zero. Equivalently, no vector in the set can be written as a combination of the others.
Put the vectors in as columns of a matrix and reduce it to row echelon form. Count the pivots. If the number of pivots equals the number of vectors, the set is independent; if it is fewer, the set is dependent.
No. The rank can never exceed the number of components, so with more vectors than components the rank must fall short of the vector count and a dependency relation always exists.
A specific set of coefficients, not all zero, that combines the vectors into the zero vector. It identifies exactly which vectors are redundant and in what proportions, which the yes-or-no verdict alone does not.
Because exact dependence is destroyed by any perturbation at all. A tolerance decides how close to zero a pivot has to be before it counts as zero, and data with rounding or measurement error needs a looser threshold than exact integers.
Never. The zero vector can be given any non-zero coefficient while every other coefficient stays at zero, which produces a valid non-trivial combination summing to zero.
For exactly n vectors in n dimensions the matrix is square, and a non-zero determinant means independent. This tool also handles rectangular cases, where no determinant exists but the rank test still applies.
This page explains a linear algebra computation for educational purposes. Results depend on the tolerance you select, and near-dependence in real data should be interpreted against the precision of the measurements it came from.
