How this calculator works
Place the vectors as rows of a matrix and reduce it. The rank counts the independent rows. If the rank equals the number of vectors, they are linearly independent. If it is smaller, at least one vector lies in the span of the others. The null space of the transposed matrix then gives the coefficients of an explicit dependence relation. In two or three dimensions, the determinant also decides the square case: a non-zero determinant means independence.
Formula
Vectors v1 to vk are independent when c1 v1 + ... + ck vk = 0 forces c1 = ... = ck = 0. Equivalently, the rank of the matrix with those rows equals k.
Worked example
The vectors 1, 0 and 0, 1 are independent because their rank is 2. The vectors 1, 2 and 2, 4 are dependent because the second is twice the first, so rank is 1. Three vectors in the plane, such as 1, 0; 0, 1; 1, 1, can never be independent.
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Frequently asked questions
Can more than n vectors be independent in n dimensions?
No. At most n vectors can be independent in n-dimensional space, because the rank cannot exceed the number of coordinates.
What does the dependence relation show?
It gives non-zero coefficients that combine to the zero vector. Rearranging it expresses one vector as a combination of the others.
How is independence related to the determinant?
For the same number of square vectors, a non-zero determinant means independence. A zero determinant means dependence.